An inverse technique for retrieving higher mode phase velocity and mantle structure

SUMMARY It is well established that the Earth’s uppermost mantle is anisotropic, but there are no clear observations of anisotropy in the deeper parts of the mantle. Surface waves are well suited to observe anisotropy since they carry information about both radial and azimuthal anisotropy. Fundamental mode surface waves, for commonly used periods up to 200 s, are sensitive to structure in the first few hundred kilometres, and therefore, do not provide information on anisotropy below. Higher mode surface waves have sensitivities that extend to and beyond the transition zone, and should thus give insight about azimuthal anisotropy at greater depths. We have measured higher mode Love and Rayleigh phase velocities using a model space search approach, which provides us with consistent relative uncertainties from measurement to measurement and from mode to mode. From these phase velocity measurements, we constructed global anisotropic phase velocity maps. Prior to inversion, we determine the optimum relative weighting for anisotropy. We present global azimuthal phase velocity maps for higher mode Rayleigh waves (up to the sixth higher mode) and Love waves (up to the fifth higher mode) with corresponding average model uncertainties. The anisotropy we derive is robust within the uncertainties for all modes. Given the ray theoretical sensitivity kernels of Rayleigh and Love wave modes, the source of anisotropy is complex, but mainly located in the asthenosphere and deeper. Our models show a good correspondence with other studies for the fundamental mode, but we have been able to achieve higher resolution.

Surface waves travelling along the subsurface interface between water and sediment are called Scholte (or generalised Rayleigh) waves. In this paper, we examine the characteristics of Scholte wave dispersion curves and their modal energy distribution through synthetic simulation tests. Analytical solutions of the dispersion equation for a homogeneous isotropic layered medium describe the propagation characteristics of the fundamental and higher modes, but do not provide information about the relative energy distributions of those wave modes. In real situations, the dispersion curves with different modes have different energies corresponding to different frequency bands. It is instructive to study and analyse the behaviours of Scholte wave dispersion curves by using joint dispersion equation solutions and numerical simulations results. Dispersion equation solutions are used to validate the numerical results, which in turn are used to assess energy distribution for both the fundamental and the higher modes. We observe that the thickness of the overlying water layer and the elastic properties of the ocean bottom play important roles in the dispersion characteristics of Scholte waves. The dispersion of the soft ocean-bottom model (whose S-wave velocity of the first solid layer is lower than the P-wave velocity of water) is dominated by the fundamental mode and appears to be less dependent on the thickness of the overlying water layer. In contrast, the dispersion of the hard ocean-bottom model contains significant energy in the higher modes. Higher modes will shift to lower frequencies and will be spatially closer (in the frequency direction) when the water layer thickness increases. The dispersion characteristics of the real data are in a good agreement with the conclusion of a soft ocean-bottom model. Our modelling approach of combining analytical solutions of the dispersion equation with numerical modelling can facilitate Scholte wave exploration in water-covered areas.

The interest of both the scientific community and the professionals towards the MASW method (Multichannel Spectral Analysis of Surface Waves) has been increasing for the last years.cIn the ’90 several researchers have realized that, when dealing with inversely dispersive sites, the MASW method based only on the fundamental mode can really cause erroneous Vs profiles, hence an erroneous seismic site characterization. When dealing with inversely dispersive sites ( i.e. sites where stiffness discontinuities exist, soft layers trapped between stiffer layers or viceversa stiff layers trapped between softer layers) higher modes of Rayleigh waves must be combined together with the fundamental mode to calculate the effective or apparent dispersion curve (Lai 1998, Roma 2001-2002-2006), in order to achieve a reliable Vs profile and a reliable seismic site characterization. It is not sufficient to calculate the numerical higher modes and use them separately for the inversion process, because it is practically impossible to distinguish the experimental higher modes from the field data in the geotechnical scale. It is well known that the apparent experimental dispersion curve that is determined from the field data is the result of a superposition of the several higher modes. In order to obtain a reliable Vs profile and hence a reliable site soil characterization not only the fundamental mode, but also the higher modes of Rayleigh have be combined when calculating the apparent numerical dispersion curve. Consider that it is not sufficient to calculate the higher modes of Rayleigh, because it is necessary to combine the higher modes with the fundamental mode to calculate the numerical apparent dispersion curve, as it is measured in field. This is due to the fact that from field data it is not possible to distinguish and define the higher modes without a large uncertainty. In this article the potentialities of a new algorithm (www.masw.it, Roma 2001) that calculates the apparent dispersion curve using both the fundamental mode and all higher modes are shown into an application to a real case.

Summary. The paper presents a method of investigating seismic surface waves in (two-dimensional) layered media involving lateral heterogeneities. The method combines forward modelling by computation of synthetic seismograms for the disturbed layering and subsequent inversion of the seismogram spectra with respect to the wave modes. In terms of ‘L region’ and ‘R region', denoting the laterally homogeneous regions of the waves before and after propagation across the disturbed region, the method is described as follows. (1) Analytical definition of the wave in the L region. A time-dependent wavelet is assumed at some surface point, and the spatial displacement field corresponding to a free plane surface wave is computed from the eigen-functions of the waveguide. (2) Numerical propagation of the wave across the laterally heterogeneous region. The general heterogeneous wave equation is approximated by a finite difference time marching scheme, where the analytical wave motion enters as initial value. (3) Inversion of seismograms ‘observed’ at a number of locations in the R region. The seismogram Fourier spectra are modelled by a linear combination of modes, contaminated by random noise. Least-squares fitting between the modelled and the observed seismogram spectra yields estimates of the ‘mode participation factors'. The method is applied to Love waves of an elastic upper crustal model composed of a layer (shear velocity β=2.5 km s-1) over a homogeneous substratum (β=3.5 km s-1). Embedded in the surface layer, of 10 km thickness, is an idealized rectangular shaped sediment basin with lateral and vertical dimensions of 15 and 5 km, and a velocity of 0.6 or 0.4 of that of the layer. The initial pulse in the form of a Ricker wavelet is defined to represent a pure fundamental mode, the wavelengths of which vary between 60 and 6 km in the frequency interval 0.05-0.42 Hz ('analysis interval'), the wavelength of maximum spectral amplitude being 15 km. Observations at a distance of 60 km beyond the sediment basin yield the following results. (1) Propagation across the basin gives rise to significant scattering of the fundamental mode energy into the higher modes. The fraction of the Love wave energy transmitted is about 50 per cent for both models (slightly higher for the model of lower velocity contrast). The transmitted Love wave energy density exhibits different behaviour between the two models. For the model of lower velocity contrast, there is an increase with frequency from about 30 to 80 per cent; for the model of higher velocity contrast, there are variations between corresponding values, without a significant frequency trend. (2) Owing to the multimodal character of the Love wave transmitted, the phase velocity estimated from the horizontal phase gradient, differs from the fundamental mode velocity by amounts of up to about 10 per cent. Moreover, phase velocity as well as the seismogram spectral amplitudes are dependent upon the horizontal coordinate. This means that it is generally inadequate to interpret phase velocity (from phase differences) or attenuation (from amplitude ratios) in corresponding subsurface structures in terms of one single mode.

One of the goals of the Interior Exploration using Seismic Investigations, Geodesy and Heat Transport (InSight) mission is to constrain the interior structure of Mars. We present a hierarchical transdimensional Bayesian approach to extract phase velocity dispersion and interior shear‐wave velocity (VS) models from a single seismogram. This method was adapted to Mars from a technique recently developed for Earth (Xu & Beghein, 2019, https://doi.org/10.1093/gji/ggz133). Monte Carlo Markov Chains seek an ensemble of one dimensional (1‐D) VS models between a source and a receiver that can explain the observed waveform. The models obtained are used to calculate the phase velocities of fundamental and higher modes at selected periods, and a subsequent analysis is performed to assess which modes were reliably measured. An advantage of our approach is that it can also fit unknown data noise, which reduces the risk of overfitting the data. In addition, uncertainties in the source parameters can be propagated, yielding more accurate model parameter uncertainties. In this study, we first present our technique and discuss the challenges stemming from using a single station to characterize both structure and the source and from the absence of a Mars reference model. We then demonstrate the method feasibility using the Mars Structure Service blind test data and our own synthetic data, which included realistic noise levels based on the noise recorded by InSight.

The axes ratio of elliptically-polarised Rayleigh-wave particle motion may provide information about shallow crustal structure beneath a seismographic station. This seldom-used parameter, often called ellipticity, or H/V, is different from the popular horizontal to vertical amplitude ratio of ambient seismic noise used for microzonation and similar studies, as we refer to deterministic measurements on Rayleigh-wave, fundamental-mode, wave groups originated by distant earthquakes. This measure is not influenced either by the source, or the propagation, and may yield robust information on crustal structure at shallower depth than group or phase velocity at the same frequency. Ellipticity is a potentially excellent tool to retrieve seismic parameters in sedimentary basins. We intend to examine feasibility and robustness of ellipticity measurements, and their application to retrieval of shallow crustal structure in northern Italy. The most striking structural element is the Po Plain sedimentary basin, but also sedimentary and crystalline rocks in the Northern Apennines and Alps. The Po Plain hides a complex system of active thrusts and folds that caused the 2012 seismic sequence. Our aim is to set a reliable system to measure and use H/V ratio to study the shallow crustal structure of this region. We implement and test automatic measurements, for 95 seismic stations in northern Italy. Comparison between observations and predictions from a reference crustal model show substantial fit, particularly for T ≈ 38s data. Discrepancy for shorter periods suggests that slight modifications of the model are needed. Analysis of synthetic and real data indicates the possibility of prograde (inverse) particle motion, but the influence of ambient noise at the predicted, short, transition periods, makes such observations less reliable. We further invert ellipticity curves to retrieve Vs profiles beneath each station, and finally build a high-resolution crustal model of the Po Plain.

Surface wave dispersion curves derived from ambient noise recordings are frequently used to invert for subsurface velocity information. Rayleigh wave ellipticities and phase velocities are exploited, and sometimes jointly inverted, for the velocity structure beneath seismic arrays. Wavelengths of surface waves become large at low frequencies and are, thus, sensitive to great depths, but provide only very smooth velocity profiles. However, sudden velocity increases in the subsurface are of particular interest to delineate the extent of reservoirs, i.e., by sub-horizontal faults or detachments, or estimate the depth of sedimentary basins.Here, we report a new approach to estimate sudden velocity increases in vertical velocity profiles using Rayleigh wave ellipticities and phase velocities. Using Kepler’s law of motion on elliptical orbits, we can theoretically delineate the frequency-dependent half-height and half-width of the energy ellipse described by Rayleigh waves.In the presence of sudden velocity increases, fundamental and first higher mode Rayleigh waves have similar phase velocities at the so-called osculation frequency. This often leads to mode misidentification that biases inversion results. We show that this osculation frequency is close to the frequency where the Rayleigh ellipticity of the fundamental mode is one, i.e., motion is circular, and the ellipticity of the first higher mode has its maximum. At this frequency, our derived relation only requires the phase velocity of the first higher mode to estimate the half-height of the ellipse, which is a very good approximation of the depth of the sudden velocity increase.To derive phase velocities and ellipticities of Rayleigh waves for synthetic three-component waveforms and real-world datasets from three sites (Weisweiler in Germany, FORGE in Utah, USA and Groningen, the Netherlands), we use three-component beamforming, which provides velocity and polarization parameters of recorded waves in short ambient noise time windows and thus can distinguish wave types and modes. From identified Rayleigh waves, we pick the phase velocity of the first higher mode at the osculation frequency directly in the beamformer plots and estimate the depth of sudden velocity increases using our new relation. No inversion scheme is needed for this approach.This approach provides more accurate depth estimates of velocity jumps than other ambient noise methods. The depth sensitivity is only limited by the inter-station distances in the array configuration and the useable frequency range. The derived depths of sudden velocity increases can be used to constrain inversion schemes for more accurate velocity models or can be used directly to map structural changes in the subsurface.

In this paper we are going to show the importance of mode identification in microtremor array analysis. The idea come from four concentric ambient noise array recordings with aperture 100 to 1000 meters, performed in southern urban area of Tehran near the shrine of Imam Khomeini. These measurements were part of a comprehensive research project with the aim of determination of deep shear wave velocity model of Tehran alluvial deposits. Using appropriate signal processing techniques, including array processing methods as well as classical and time-frequency horizontal/vertical spectral ratio, the dispersion curves of surface waves, fundamental resonance frequency and Ellipticity of Rayleigh waves, were extracted. In the final step, the shear wave velocity profile of the site was determined by joint inversion of all of these attributes. The results show 2 different energetic trends in dispersion curves, for arrays of aperture 200 and 400 meters that one of them is coincide with 100m aperture array. For array with aperture 1000m any clear trend of energy could be observed because of deficiency of energy in low frequency. The inversion of data obtained by 100m aperture array alone, assuming the dispersion curve as fundamental mode (a common procedure in urban area) result in shear wave velocity that is not match with existing geological information. Performing the inversion, assuming 2 energetic trends, observed for larger arrays; one as fundamental mode and another as mode 1 of Rayleigh waves, can modify significantly the shear wave velocity profile in accordance with existing geological and geotechnical information. This study show the importance of extracting of correct dispersion curves with detecting fundamental and higher modes, using array measurement with various aperture at one place to obtain more realistic shear wave velocity profile.

Summary. The array stacking method (Nolet 1975) is extended to include information from the source mechanism by adding a ‘seismogram’ constructed from theoretical phase and amplitude excitations. If the location of the everit is favourable, the effective size of the array can be significantly increased in this way, which is desirable for better resolution of higher mode dispersion. By putting the origin at the earthquake location in the stacking method it is possible to utilize the phase of the array stack as well as the amplitude as a criterion of phase velocity across the array. Also, using events close to the array of stations should minimize adverse effects of propagation across tectonic boundaries. Synthetic seismograms of multimode surface waves were processed in order to find out the expected errors in the staclung method. In the presence of noise or interference between modes, it is shown that the errors in phase velocity from the phase criterion are relatively independent of the errors in the phase velocity from the peak amplitude criterion. The array stacking method is applied to an array composed of western US WWSSN stations recording two events in southern Mexico. This array lies within the western North America Cordilleran Belt, a large region characterized by low seismic wave velocity and a high level of tectonic activity. The fundamental mode and five higher mode Rayleigh wave dispersion curves for this region are derived from stacking the data. These have low velocity compared with average continental structure, but agree with dispersion curves computed from published average western US structures.

<p>It has been widely recognized that the cross-correlation function (CCF) of ambient noise data recorded at two seismic stations approximates to the part of Green’s Function between these two stations. Theoretically, the CCF should include the higher modes, apart from the fundamental mode. However, currently well-known and mature methods that can extract dispersion curves are not pretty proficient in extracting higher modes. Fortunately, our newly proposing method, the Frequency-Bessel Transform Method (F-J Method), has presented its obvious advantage in extracting higher modes. This study applied F-J method to seismic ambient noise data for the east of South China, including Jiangnan Orogen and South China Fold System. We have acquired higher modes, not to mention the fundamental mode with wider frequency than previous studies. Combining both fundamental mode and higher modes, we used L-BFGS inversion method to inverse and acquire more accurate crustal and upper-mantle structure than previous studies only adopting fundamental mode for the east of South China. As shown in this study for the east of South China, we can use F-J method  to conveniently and precisely extract multimodes from ambient noise data and thus add more constrains for inversion results, which can significantly improve the preciseness of imaging crustal and upper-mantle structure.</p>

SUMMARY Several filtering techniques have been used to remove the effects of multipathing and modal contamination, and to isolate the fundamental mode from Rayleigh wavetrains. Group velocity data are obtained by means of the multiple-filter technique. A time-variable filter has allowed the influence of noise as well as the interference produced by higher modes to be removed. Multiple filtering is then used again to compute group velocities at each station. the interstation group velocity for the fundamental mode Rayleigh wave is estimated according to the velocities at two stations. Frequency-domain Wiener deconvolution is used to compute the phase velocity between two stations. the well-known three-station method is applied to correct the distances travelled by the waves across the array and therefore to determine interstation phase and group velocities in a more accurate manner. On the other hand, lateral refraction at the Atlantic continental edge of the Peninsula is also studied. Phase velocities are corrected for the anelastic effect. Inversion of the interstation Rayleigh wave phase velocities is then made in accordance with generalized inversion theory to obtain theoretical 2-D layered earth models. In this paper, these methods are applied to Rayleigh waves generated by teleseismic events propagating across the Iberian Peninsula and recorded at WWSSN stations. As a consequence, new and principal features for the Iberian lithosphere-asthenosphere system are obtained. A very interesting feature of the the Iberian lithosphere was found-a low-velocity layer directly under the Moho, between 39 and 64 km depth, with shear velocities ranging from 4.12 to 4.37 km s-1. the Iberian asthenosphere, which lies between 100 and 180km depth, is not an homogeneous layer and shows a negative velocity gradient from top to bottom together with a sudden increase in shear velocity beneath the low-velocity zone.

SUMMARY A fully automated method for obtaining multimode phase speed measurements from a single seismogram has been developed and applied to a large data set of three-component longperiod seismograms in North America, constructing high-resolution phase speed maps on a continental scale. The method of our phase speed estimation is based on a fully non-linear waveform inversion by Yoshizawa & Kennett working with a global search method (the Neighbourhood Algorithm). The entire process of waveform fitting and the evaluation of the estimated phase speed have been fully automated employing several empirical quantitative measures, assessing the quality of waveform fit and the relative contributions of each mode in a chosen time window. The measured phase speed data undergo automatic screening for quality control, comprising the threshold evaluation of their reliability and outlier detection and removal. This new automated method has been applied to a large data set recorded at North American stations, including the latest transportable stations of USArray. Using long-period three-component seismograms recorded during the past eight years, we have successfully retrieved large numbers of regional surface wave paths, including over 20000 paths for the fundamental-mode Rayleigh waves over a wide range of frequencies, and over 10000 paths for the higher mode Rayleigh as well as the fundamental-mode Love waves. The consistentresultsoftheautomatedmeasurementproceduresuggestthatthemethodworkswell at regional distances, allowing us to perform a high-resolution mapping of multimode phase speeds in North America. The results of the automated waveform analysis also indicate some intrinsic limitations in the higher mode phase speed measurements from a single seismogram particularly in the short period range, mainly due to the overlapping of higher mode arrivals as well as coupling between mode branches. Despite such an innate difficulty in the higher mode dispersion measurements, the automated method allows us to construct reliable multimode phase speed maps. The current data set of ray paths is significantly biased towards the western half of North America, resulting in non-uniform horizontal resolution across the continent. This issue will be resolved by the future migration of the USArray stations to cover the central and eastern United States. The new automated method can be a useful tool for high-resolution mapping of regional 3-D shear wave structure including possible anisotropy.

SUMMARY In most tomographic inversion of surface wave data, the long-period seismograms are first interpreted in terms of dispersion and/or attenuation curves before performing an inversion in terms of laterally and depth-varying properties. An alternative to this approach is to perform a direct waveform inversion or, as in Cara & Levěque, (1987), to use another set of secondary observables built up from the seismograms. In this paper, we systematically test with actual Rayleigh wave records this latter technique by considering laterally homogeneous models. We use for this purpose a set of recent long-period digital records from the Geoscope station Dumont d'Urville, Antarctica, for surficial events in the south Indian Ocean and the southeast Pacific Ocean, and for intermediate-depth events in the Vanuatu and Kermadec trenches. In addition to the obvious advantages of being able to deal with situations where overtones are present in the seismogram, it is found that the waveform inversion procedure allows us to reach a better depth resolution than in classical inversion of dispersion curves, even when only the fundamental mode is present in the seismogram. Quite good resolution is obtained to depth as large as 300km for S velocity when using surficial events located at a few thousand kilometres from the station, while classical surface wave studies do not allow us to resolve S velocity at depth larger than 150 km for these events. When intermediate-depth events are used at distances of about 5000 km, the presence of overtones in the signal allows us to get resolution to depth as large as 600 km for S velocity. Poorer resolution is obtained in both situations for the attenuation factor. S velocity appears furthermore to be more robust than attenuation at depth where good resolution is achieved. Due to the great sensitivity of surface wave amplitude to departure from the assumption of lateral homogeneity, more sophisticated direct modelling would be required to get more confidence in the inverted attenuation models.

We show that higher modes are an important component of high-frequency Rayleigh waves in the cross-correlations over sedimentary basins. The particle motions provide a good test for distinguishing and separating the fundamental from the first higher mode, with the fundamental mode having retrograde and the first higher mode having prograde motion in the 1–10 s period of interest. The basement depth controls the cut-off period of the first higher mode, which coincides with a rapid increase (over period) in the particle-motion ellipticity or H/V ratio of the fundamental mode. The strong higher mode we observed is not only due to the low-velocity sedimentary layer but also due to the noise sources with significant radial component such as the basin edge scattering. It is important to correctly identify the mode order when inverting the dispersion curves because misidentifying the higher mode as fundamental will lead to an anomalous high V_(SV) velocity.

Properties of a vibrating annular membrane are analyzed mathematically and numerically to investigate the relative harmonicity of the higher-order modes compared with the fundamental mode. The harmonicity depends on the zeros of a cross-product of Bessel functions. For a ratio of external-to-internal diameter up to about 1.2, the higher modes have frequencies which are very nearly exact integer multiples of the fundamental frequency. For circularly symmetric modes, the ratio of all higher radial mode frequencies to exact harmonic values is less than 1.0006 (a logarithmic cent), a ratio imperceptible to a listener. Angle-dependent modes also give good harmonicity, especially if some lower radial modes are suppressed. Thus the modes of an annular membrane whose external-to-internal diameter ratio is less than 1.2 are much more harmonic than those of the full circular membrane of a conventional drum. The theoretical harmonicity is at least as good as reported for stringed melodic instruments (piano, violin). The analysis suggests characteristics for construction of an annular drum-head which is supported on concentric rims of two cylinders; to sound the drum, a braced hoop might be used to excite radial modes, or strikers with grooved faces might be used to excite angular modes without exciting lower radial modes.