scholarly journals Path-Following Methods for Calculating Linear Surface Wave Dispersion Relations on Vertical Shear Flows

Water Waves ◽  
2020 ◽  
Vol 2 (2) ◽  
pp. 381-414 ◽  
Author(s):  
Peter Maxwell ◽  
Simen Å. Ellingsen
Keyword(s):  
Dispersion Relation ◽  
Wave Vector ◽  
Path Following ◽  
Wave Dispersion ◽  
Vertical Shear ◽  
Companion Matrix ◽  
Surface Boundary ◽  
Setup Cost ◽  
Surface Wave Dispersion ◽  
Path Following Methods

Abstract The path-following scheme in Loisel and Maxwell (SIAM J Matrix Anal Appl 39(4):1726–1749, 2018) is adapted to efficiently calculate the dispersion relation curve for linear surface waves on an arbitrary vertical shear current. This is equivalent to solving the Rayleigh stability equation with linearized free-surface boundary condition for each sought point on the curve. Taking advantage of the analyticity of the dispersion relation, a path-following or continuation approach is adopted. The problem is discretized using a collocation scheme, parametrized along either a radial or angular path in the wave vector plane, and differentiated to yield a system of ODEs. After an initial eigenproblem solve using QZ decomposition, numerical integration proceeds along the curve using linear solves as the Runge–Kutta $$F(\cdot )$$ F ( · ) function; thus, many QZ decompositions on a size 2N companion matrix are exchanged for one QZ decomposition and a small number of linear solves on a size N matrix. A piecewise interpolant provides dense output. The integration represents a nominal setup cost whereafter very many points can be computed at negligible cost whilst preserving high accuracy. Furthermore, a two-dimensional interpolant suitable for scattered data query points in the wave vector plane is described. Finally, a comparison is made with existing numerical methods for this problem, revealing that the path-following scheme is the most competitive algorithm for this problem whenever calculating more than circa 1,000 data points or relative normwise accuracy better than $$10^{-4}$$ 10 - 4 is sought.

A surface wave dispersion relation for non-local media

1978 ◽  
Vol 28 (11) ◽  
pp. 927-929 ◽  
Author(s):  
Deva N. Pattanayak ◽  
Joseph L. Birman
Keyword(s):  
Surface Wave ◽  
Dispersion Relation ◽  
Wave Dispersion ◽  
Surface Wave Dispersion ◽  
Local Media ◽  
Non Local

Improved Surface Wave Dispersion Models, Amplitude Measurements and Azimuth Estimates

2005 ◽  
Author(s):  
Jeffry L. Stevens ◽  
David A. Adams ◽  
G. E. Baker ◽  
Mariana G. Eneva ◽  
Heming Xu
Keyword(s):  
Surface Wave ◽  
Wave Dispersion ◽  
Dispersion Models ◽  
Surface Wave Dispersion

scholarly journals Imaging the subsurface using induced seismicity and ambient noise: 3-D tomographic Monte Carlo joint inversion of earthquake body wave traveltimes and surface wave dispersion

2020 ◽  
Vol 222 (3) ◽  
pp. 1639-1655
Author(s):  
Xin Zhang ◽  
Corinna Roy ◽  
Andrew Curtis ◽  
Andy Nowacki ◽  
Brian Baptie
Keyword(s):  
Surface Wave ◽  
Ambient Noise ◽  
Velocity Structure ◽  
Joint Inversion ◽  
Body Wave ◽  
Source Parameters ◽  
Wave Dispersion ◽  
Inversion Method ◽  
Surface Wave Dispersion ◽  
Wave Data

SUMMARY Seismic body wave traveltime tomography and surface wave dispersion tomography have been used widely to characterize earthquakes and to study the subsurface structure of the Earth. Since these types of problem are often significantly non-linear and have non-unique solutions, Markov chain Monte Carlo methods have been used to find probabilistic solutions. Body and surface wave data are usually inverted separately to produce independent velocity models. However, body wave tomography is generally sensitive to structure around the subvolume in which earthquakes occur and produces limited resolution in the shallower Earth, whereas surface wave tomography is often sensitive to shallower structure. To better estimate subsurface properties, we therefore jointly invert for the seismic velocity structure and earthquake locations using body and surface wave data simultaneously. We apply the new joint inversion method to a mining site in the United Kingdom at which induced seismicity occurred and was recorded on a small local network of stations, and where ambient noise recordings are available from the same stations. The ambient noise is processed to obtain inter-receiver surface wave dispersion measurements which are inverted jointly with body wave arrival times from local earthquakes. The results show that by using both types of data, the earthquake source parameters and the velocity structure can be better constrained than in independent inversions. To further understand and interpret the results, we conduct synthetic tests to compare the results from body wave inversion and joint inversion. The results show that trade-offs between source parameters and velocities appear to bias results if only body wave data are used, but this issue is largely resolved by using the joint inversion method. Thus the use of ambient seismic noise and our fully non-linear inversion provides a valuable, improved method to image the subsurface velocity and seismicity.

scholarly journals Reliable workflow for inversion of seismic receiver function and surface wave dispersion data: a “13 BB Star” case study

2019 ◽  
Vol 24 (1) ◽  
pp. 101-120
Author(s):  
Kajetan Chrapkiewicz ◽  
Monika Wilde-Piórko ◽  
Marcin Polkowski ◽  
Marek Grad
Keyword(s):  
Surface Wave ◽  
Inverse Problems ◽  
Receiver Function ◽  
Joint Inversion ◽  
Wave Dispersion ◽  
Receiver Functions ◽  
P Wave ◽  
Surface Wave Dispersion ◽  
Phase Velocity Dispersion ◽  
Wide Range

AbstractNon-linear inverse problems arising in seismology are usually addressed either by linearization or by Monte Carlo methods. Neither approach is flawless. The former needs an accurate starting model; the latter is computationally intensive. Both require careful tuning of inversion parameters. An additional challenge is posed by joint inversion of data of different sensitivities and noise levels such as receiver functions and surface wave dispersion curves. We propose a generic workflow that combines advantages of both methods by endowing the linearized approach with an ensemble of homogeneous starting models. It successfully addresses several fundamental issues inherent in a wide range of inverse problems, such as trapping by local minima, exploitation of a priori knowledge, choice of a model depth, proper weighting of data sets characterized by different uncertainties, and credibility of final models. Some of them are tackled with the aid of novel 1D checkerboard tests—an intuitive and feasible addition to the resolution matrix. We applied our workflow to study the south-western margin of the East European Craton. Rayleigh wave phase velocity dispersion and P-wave receiver function data were gathered in the passive seismic experiment “13 BB Star” (2013–2016) in the area of the crust recognized by previous borehole and refraction surveys. Final models of S-wave velocity down to 300 km depth beneath the array are characterized by proximity in the parameter space and very good data fit. The maximum value in the mantle is higher by 0.1–0.2 km/s than reported for other cratons.

scholarly journals Surface wave dispersion measurements from ambient seismic noise analysis in Italy

2010 ◽  
Vol 180 (3) ◽  
pp. 1242-1252 ◽  
Author(s):  
Hongyi Li ◽  
Fabrizio Bernardi ◽  
Alberto Michelini
Keyword(s):  
Surface Wave ◽  
Seismic Noise ◽  
Noise Analysis ◽  
Wave Dispersion ◽  
Ambient Seismic Noise ◽  
Surface Wave Dispersion

Spin wave dispersion relation in the ferromagnetic and helical phases of the disordered alloy Tb0.5Ho0.5

1978 ◽  
Vol 11 (23) ◽  
pp. 4793-4802 ◽  
Author(s):  
P Hendy ◽  
E W Lee ◽  
M W Stringfellow ◽  
M T Hutchings
Keyword(s):  
Dispersion Relation ◽  
Spin Wave ◽  
Wave Dispersion ◽  
Spin Wave Dispersion ◽  
Disordered Alloy

DISPERSION IN SEISMIC SURFACE WAVES

Geophysics ◽  
1951 ◽  
Vol 16 (1) ◽  
pp. 63-80 ◽  
Author(s):  
Milton B. Dobrin
Keyword(s):  
Surface Waves ◽  
Water Waves ◽  
Seismic Refraction ◽  
Wave Theory ◽  
Wave Dispersion ◽  
Surface Zone ◽  
Surface Wave Dispersion ◽  
Near Surface ◽  
Arrival Times ◽  
Ground Roll

A non‐mathematical summary is presented of the published theories and observations on dispersion, i.e., variation of velocity with frequency, in surface waves from earthquakes and in waterborne waves from shallow‐water explosions. Two further instances are cited in which dispersion theory has been used in analyzing seismic data. In the seismic refraction survey of Bikini Atoll, information on the first 400 feet of sediments below the lagoon bottom could not be obtained from ground wave first arrival times because shot‐detector distances were too great. Dispersion in the water waves, however, gave data on speed variations in the bottom sediments which made possible inferences on the recent geological history of the atoll. Recent systematic observations on ground roll from explosions in shot holes have shown dispersion in the surface waves which is similar in many ways to that observed in Rayleigh waves from distant earthquakes. Classical wave theory attributes Rayleigh wave dispersion to the modification of the waves by a surface layer. In the case of earthquakes, this layer is the earth’s crust. In the case of waves from shot‐holes, it is the low‐speed weathered zone. A comparison of observed ground roll dispersion with theory shows qualitative agreement, but it brings out discrepancies attributable to the fact that neither the theory for liquids nor for conventional solids applies exactly to unconsolidated near‐surface rocks. Additional experimental and theoretical study of this type of surface wave dispersion may provide useful information on the properties of the surface zone and add to our knowledge of the mechanism by which ground roll is generated in seismic shooting.

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