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![Demonstration of conventional wave gradiometry for synthetic data, as analyzed by the gradiometer shown in Figure 1. In the top half of the figure, the synthetic data consists of two arrivals, both with different propagation directions and separated in time (termed S1 in the text). In the bottom half of the figure, the synthetic data consists of two arrivals, both with different propagation directions but overlapping in time (S2). For all the panels on the left, conventional wave gradiometry is applied to the unfiltered, synthetic signals to resolve back azimuth (θ), magnitude slowness (|p|), and geometrical spreading (Ar). For the middle panels, a two-pole Butterworth high-cut filter was applied to the data prior to gradiometric analysis, and for the panels on the right, a two-pole Butterworth low-cut filter was applied to the data prior to gradiometric analysis.]()
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![(a) Multiwavelet gradiometry on synthetic data (no noise). The top panel shows the synthetic data as recorded by the master station. Two waves, separated in time, traverse the array: the first wave has a central period of approximately 0.5 s and traverses the array with a back azimuth of 45°. The second wave has a central period of 0.25 s and has a back azimuth of 225°. The second panel shows the resolved back azimuth as a function of time and period. The third panel shows the average resolved magnitude slowness, and the bottom panel shows the average resolved geometrical spreading. (b) Multiwavelet gradiometry on synthetic data (no noise). For this figure, three constant-period slices were taken of the results shown in (a). The central period is indicated at the top of each column of graphs. The top row of graphs shows the resolved back azimuth. The black line is the mean resolved back azimuth, and the gray lines indicate the 95% confidence intervals. The second row of graphs show the resolved magnitude slowness, as well as the 95% confidence intervals. The third row of graphs shows the resolved geometrical spreading, which also displays erratic behavior in the presence of wave interference. The bottom three rows are similar to the top three rows except that the results were determined using conventional wave gradiometry and are shown for comparison.]()
![(a) Multiwavelet gradiometry on synthetic data (no noise). The top panel shows the synthetic data as recorded by the master station. Two waves, separated in time, traverse the array: the first wave has a central period of approximately 0.5 s and traverses the array with a back azimuth of 45°. The second wave has a central period of 0.25 s and has a back azimuth of 225°. The second panel shows the resolved back azimuth as a function of time and period. The third panel shows the average resolved magnitude slowness, and the bottom panel shows the average resolved geometrical spreading. (b) Multiwavelet gradiometry on synthetic data (no noise). For this figure, three constant-period slices were taken of the results shown in (a). The central period is indicated at the top of each column of graphs. The top row of graphs shows the resolved back azimuth. The black line is the mean resolved back azimuth, and the gray lines indicate the 95% confidence intervals. The second row of graphs show the resolved magnitude slowness, as well as the 95% confidence intervals. The third row of graphs shows the resolved geometrical spreading, which also displays erratic behavior in the presence of wave interference. The bottom three rows are similar to the top three rows except that the results were determined using conventional wave gradiometry and are shown for comparison.]()
![Distributions of (a) amplitudes, (b) slowness vectors, (c) geometrical spreading terms, and (d) radiation pattern terms calculated using wave gradiometry for the S‐net OBPG and the SORATENA barometer arrays for the time interval of atmospheric Lamb waves. Circles in panel (a) represent the locations of the S‐net OBPGs and the SORATENA barometers. The upper color scale in panel (a) is for the barometer array, that is, land area, and the lower color scale is for the OBPG array, that is, the offshore area. Gray contours plotted as each 50 m/s in panels (c) and (d) show the propagation velocities predicted by linear long‐wave theory and bathymetry. The color version of this figure is available only in the electronic edition.]()
![(a) Assumed bathymetry around the S‐net region used for the case of simple bathymetry, (b) snapshot of the 2D amplitude distribution of synthesized tsunami, and distribution of (c) amplitudes, (d) slowness vectors, (e) geometrical spreading terms, and (f) radiation pattern terms calculated using wave gradiometry and the synthetics with the simple bathymetry. Circles in panels (a–c) represent locations of the S‐net OBPGs. Gray contours plotted as each 200 m in panel (a) show bathymetry and plotted as each 10 m/s in panels (e) and (f) show the propagation velocities predicted by linear long‐wave theory and bathymetry shown in panel (a). The color version of this figure is available only in the electronic edition.]()
![Experiment 3 scatter plots for data from shot point 1 showing number of time‐shifted waveforms averaged together to obtain the center‐station waveform that was used in the wave‐gradiometry data analysis versus (a) P‐wave slowness, (b) P‐wave azimuth, (c) surface‐wave slowness, and (d) surface‐wave azimuth. As the number of center stations increased (increasing x), the range of slowness and azimuth values decreased. The scatter plots of (e) original RHALL surface‐wave slowness estimates and (f) azimuth estimates as black dots, and the same data are processed using the center‐station correlation method (gray dots).]()
![(a) Bathymetry of the ETOPO1 topography model (NOAA National Centers for Environmental Information, 2009) and corresponding ray paths of tsunamis from the Hunga Tonga–Hunga Ha’apai volcano (black lines), (b) 2D amplitude distribution of the synthesized tsunami with the ETOPO1 topography model, and distributions of (c) amplitudes, (d) slowness vectors, (e) geometrical spreading terms, and (f) radiation pattern terms calculated using wave gradiometry at the time the synthesized tsunami waves initially propagated through the analysis region. Circles in panels (a–c) represent the location of the S‐net OBPGs. Gray contours plotted as each 50 m/s in panels (e,f) show the propagation velocities predicted by linear long‐wave theory and bathymetry. The color version of this figure is available only in the electronic edition.]()
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Journal Article
Propagation Characteristics of Tsunamis in an Atmosphere‐Ocean Coupled System Revealed by Wave Gradiometry: The 2022 Hunga Tonga–Hunga Ha’apai Eruption Case Available to Purchase
Journal: Seismological Research Letters
Publisher: Seismological Society of America
Published: 29 January 2025
Seismological Research Letters (2025) 96 (2A): 744–757.
... directions and velocities. The wave gradiometry analyses provided several interesting results. Coherent sea‐surface disturbances accompanied by the atmospheric Lamb and Pekeris waves propagated mainly in directions and velocities that reflected the bathymetry, that is, they propagated as tsunamis, whereas...
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Includes: Supplemental Content
Journal Article
Ocean‐Wave Gradiometry: Visualizing and Extracting Propagation Features of the 15 January 2022 Tsunami Wavefield with Dense Ocean‐Bottom Pressure Gauge Arrays Available to Purchase
Journal: Seismological Research Letters
Publisher: Seismological Society of America
Published: 23 December 2022
Seismological Research Letters (2023) 94 (2A): 626–636.
... have affected the propagation of the sea‐surface height changes, we investigated the propagation properties of the sea‐bottom pressure disturbances recorded by the OBPG arrays around Japan using wave gradiometry. We found that the leading pressure disturbances propagated from southeast to northwest...
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Includes: Supplemental Content
Journal Article
Small‐Scale Array Experiments in Seismic‐Wave Gradiometry Available to Purchase
Journal: Seismological Research Letters
Publisher: Seismological Society of America
Published: 10 August 2016
Seismological Research Letters (2016) 87 (5): 1091–1103.
... is given by (17) and the radiation pattern is represented as (18) There are two experiments presented in this article that use the 2D seismic‐wave gradiometry formalism. The geometrical spreading and radiation pattern are the products of this method, but we do not address them...
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Journal Article
Vertical seismic wave gradiometry: Application at the San Andreas Fault Observatory at Depth Available to Purchase
Journal: Geophysics
Publisher: Society of Exploration Geophysicists
Published: 12 April 2016
Geophysics (2016) 81 (3): D233–D243.
...Charles A. Langston; Mehari Melak Ayele ABSTRACT We have developed vertical seismic wave gradiometry (VSWG) to estimate velocity, impedance, and attenuation structure in the vicinity of boreholes using borehole array waveforms of check-shot explosions near the borehole head. We have extended wave...
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Journal Article
The Effects of Measurement Uncertainties in Seismic-Wave Gradiometry Available to Purchase
Publisher: Seismological Society of America
Published: 06 October 2015
Bulletin of the Seismological Society of America (2015) 105 (6): 3143–3155.
...Christian Poppeliers; Elizabeth V. Evans Abstract In seismic-wave gradiometry, the spatial gradient and the time derivative of the measured wavefield are used to estimate the slowness vector and changes in the radiation pattern. When using a seismic array to estimate the seismic spatial derivatives...
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Journal Article
Three‐Dimensional Seismic‐Wave Gradiometry for Scalar Waves Available to Purchase
Publisher: Seismological Society of America
Published: 01 August 2013
Bulletin of the Seismological Society of America (2013) 103 (4): 2151–2160.
...Christian Poppeliers; Predrag Punoševac; Tammy Bell Abstract Wave gradiometry relates the spatial gradients of a wavefield to its velocity and radiation patterns through two spatial coefficients for any dimension. One coefficient gives the slowness of the wave in any given dimension, and the other...
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Journal Article
Three‐Dimensional Wave Gradiometry for Polarized Seismic Waves Available to Purchase
Publisher: Seismological Society of America
Published: 01 August 2013
Bulletin of the Seismological Society of America (2013) 103 (4): 2161–2172.
...Christian Poppeliers; Predrag Punoševac Abstract We expand the theoretical development of seismic‐wave gradiometry to 3D using the polarized wavefield. First, we develop a map that relates the Cartesian spatial derivatives to the spatial derivatives in spherical coordinates. We then develop a set...
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Journal Article
Multiwavelet Seismic-Wave Gradiometry Available to Purchase
Publisher: Seismological Society of America
Published: 01 October 2011
Bulletin of the Seismological Society of America (2011) 101 (5): 2108–2121.
...Christian Poppeliers Abstract A new technique to determine uncertainty estimates for wave parameters obtained by seismic-wave gradiometry is established using the multiwavelet transform. Wave gradiometry uses spatial gradients as measured by a small-scale seismic array to estimate the wave...
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Journal Article
Seismic Wave Gradiometry Using the Wavelet Transform: Application to the Analysis of Complex Surface Waves Recorded at the Glendora Array, Sullivan, Indiana, USA Available to Purchase
Publisher: Seismological Society of America
Published: 01 June 2010
Bulletin of the Seismological Society of America (2010) 100 (3): 1211–1224.
...Christian Poppeliers Abstract This article describes a new development in seismic wave gradiometry. The work follows previous work on time-domain wave gradiometry but proposes applying a wavelet transform to the data prior to wave gradiometric analysis. The result is a more complete picture of wave...
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Journal Article
Wave Gradiometry in the Time Domain Available to Purchase
Publisher: Seismological Society of America
Published: 01 June 2007
Bulletin of the Seismological Society of America (2007) 97 (3): 926–933.
...Charles A. Langston Abstract A time-domain approach for solving for the change in geometrical spreading and horizontal wave slowness in wave gradiometry is presented based on the use of the analytic signal. The horizontal displacement gradient of a wave is linearly related to the displacement...
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Journal Article
Wave Gradiometry in Two Dimensions Available to Purchase
Publisher: Seismological Society of America
Published: 01 April 2007
Bulletin of the Seismological Society of America (2007) 97 (2): 401–416.
... application of gradiometry for three spatial dimensions. In its simplest form, a plane wave can be represented by 55 Differentiation of this equation in all three coordinates yields three 1D sets of coefficients to determine the slowness vector. Inclusion of geometrical spreading and/or radiation pattern...
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Journal Article
Seismic-Wave Strain, Rotation, and Gradiometry for the 4 March 2008 TAIGER Explosions Available to Purchase
Publisher: Seismological Society of America
Published: 01 May 2009
Bulletin of the Seismological Society of America (2009) 99 (2B): 1287–1301.
... and rotation about the vertical axis have equal amplitudes and suggest significant wave scattering within the confines of the river valley where the experiment was performed and/or significant departure from an axisymmetric explosion source. Gradiometry shows that the P wave arrives at the array 35° off...
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Image
Demonstration of conventional wave gradiometry for synthetic data, as analy... Available to Purchase
in Seismic Wave Gradiometry Using the Wavelet Transform: Application to the Analysis of Complex Surface Waves Recorded at the Glendora Array, Sullivan, Indiana, USA
> Bulletin of the Seismological Society of America
Published: 01 June 2010
Figure 2. Demonstration of conventional wave gradiometry for synthetic data, as analyzed by the gradiometer shown in Figure 1 . In the top half of the figure, the synthetic data consists of two arrivals, both with different propagation directions and separated in time (termed S1 in the text
Journal Article
An Optimum 2D Seismic‐Wavefield Reconstruction in Densely and Nonuniformly Distributed Stations: The Metropolitan Seismic Observation Network in Japan Available to Purchase
Journal: Seismological Research Letters
Publisher: Seismological Society of America
Published: 03 February 2021
Seismological Research Letters (2021) 92 (3): 2015–2027.
...Takahiro Shiina; Takuto Maeda; Masayuki Kano; Aitaro Kato; Naoshi Hirata Abstract We propose an optimization method for applying the seismic‐wave gradiometry (SWG) method to a dense seismic station network consisting of nonuniformly distributed seismographs. As a nonuniformly distributed station...
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Includes: Supplemental Content
Image
(a) Multiwavelet gradiometry on synthetic data (no noise). The top panel sh... Available to Purchase
Published: 01 October 2011
Figure 3. (a) Multiwavelet gradiometry on synthetic data (no noise). The top panel shows the synthetic data as recorded by the master station. Two waves, separated in time, traverse the array: the first wave has a central period of approximately 0.5 s and traverses the array with a back azimuth
Image
(a) Multiwavelet gradiometry on synthetic data (no noise). The top panel sh... Available to Purchase
Published: 01 October 2011
Figure 3. (a) Multiwavelet gradiometry on synthetic data (no noise). The top panel shows the synthetic data as recorded by the master station. Two waves, separated in time, traverse the array: the first wave has a central period of approximately 0.5 s and traverses the array with a back azimuth
Image
Distributions of (a) amplitudes, (b) slowness vectors, (c) geometrical spre... Available to Purchase
in Propagation Characteristics of Tsunamis in an Atmosphere‐Ocean Coupled System Revealed by Wave Gradiometry: The 2022 Hunga Tonga–Hunga Ha’apai Eruption Case
> Seismological Research Letters
Published: 29 January 2025
Figure 6. Distributions of (a) amplitudes, (b) slowness vectors, (c) geometrical spreading terms, and (d) radiation pattern terms calculated using wave gradiometry for the S‐net OBPG and the SORATENA barometer arrays for the time interval of atmospheric Lamb waves. Circles in panel (a) represent
Image
(a) Assumed bathymetry around the S‐net region used for the case of simple ... Available to Purchase
in Propagation Characteristics of Tsunamis in an Atmosphere‐Ocean Coupled System Revealed by Wave Gradiometry: The 2022 Hunga Tonga–Hunga Ha’apai Eruption Case
> Seismological Research Letters
Published: 29 January 2025
calculated using wave gradiometry and the synthetics with the simple bathymetry. Circles in panels (a–c) represent locations of the S‐net OBPGs. Gray contours plotted as each 200 m in panel (a) show bathymetry and plotted as each 10 m/s in panels (e) and (f) show the propagation velocities predicted
Image
Experiment 3 scatter plots for data from shot point 1 showing number of tim... Available to Purchase
Published: 10 August 2016
Figure 10. Experiment 3 scatter plots for data from shot point 1 showing number of time‐shifted waveforms averaged together to obtain the center‐station waveform that was used in the wave‐gradiometry data analysis versus (a) P ‐wave slowness, (b) P ‐wave azimuth, (c) surface‐wave slowness
Image
(a) Bathymetry of the ETOPO1 topography model ( NOAA National Centers for E... Available to Purchase
in Propagation Characteristics of Tsunamis in an Atmosphere‐Ocean Coupled System Revealed by Wave Gradiometry: The 2022 Hunga Tonga–Hunga Ha’apai Eruption Case
> Seismological Research Letters
Published: 29 January 2025
topography model, and distributions of (c) amplitudes, (d) slowness vectors, (e) geometrical spreading terms, and (f) radiation pattern terms calculated using wave gradiometry at the time the synthesized tsunami waves initially propagated through the analysis region. Circles in panels (a–c) represent
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