Reconstruction of a 2D seismic wavefield by seismic gradiometry
 Takuto Maeda^{1}Email authorView ORCID ID profile,
 Kiwamu Nishida^{1},
 Ryota Takagi^{2} and
 Kazushige Obara^{1}
DOI: 10.1186/s4064501601074
© The Author(s). 2016
Received: 22 May 2016
Accepted: 20 September 2016
Published: 10 October 2016
Abstract
We reconstructed a 2D seismic wavefield and obtained its propagation properties by using the seismic gradiometry method together with dense observations of the Hinet seismograph network in Japan. The seismic gradiometry method estimates the wave amplitude and its spatial derivative coefficients at any location from a discrete station record by using a Taylor series approximation. From the spatial derivatives in horizontal directions, the properties of a propagating wave packet, including the arrival direction, slowness, geometrical spreading, and radiation pattern can be obtained. In addition, by using spatial derivatives together with freesurface boundary conditions, the 2D vector elastic wavefield can be decomposed into divergence and rotation components. First, as a feasibility test, we performed an analysis with a synthetic seismogram dataset computed by a numerical simulation for a realistic 3D medium and the actual Hinet station layout. We confirmed that the wave amplitude and its spatial derivatives were very wellreproduced for period bands longer than 25 s. Applications to a real large earthquake showed that the amplitude and phase of the wavefield were well reconstructed, along with slowness vector. The slowness of the reconstructed wavefield showed a clear contrast between body and surface waves and regional nongreatcirclepath wave propagation, possibly owing to scattering. Slowness vectors together with divergence and rotation decomposition are expected to be useful for determining constituents of observed wavefields in inhomogeneous media.
Keywords
Seismic gradiometry Array signal processing Surface waves Scattering Numerical simulationIntroduction
Observations of longperiod seismic waves by dense arrays capture seismic ground motion directly as spatially propagating waves, rather than as individual traces at discrete stations. Thus, seismic waves propagating across arrays have been visualized for various arrays (e.g., Obara et al. 2005; Sheldrake et al. 2002; Trabant et al. 2012). Such visualization is useful not only for educational or outreach purposes but also for capturing characteristic wave propagation. By treating a seismic wave as a spatially varying field, Maeda et al. (2014) detected a new sort of longperiod scattered wave that originated from reverberation in the seawater column within a deep trench.
In this study, we applied a relatively new technique for treating a large set of array data, called seismic gradiometry, to reconstruct and characterize a seismic wavefield observed by the Hinet array. The seismic gradiometry technique (Langston 2007a) was originally developed as a method to measure the spatial gradient of seismic waves observed by dense arrays. Langston (2007a) showed that the slowness could be directly estimated from the spatial gradient. Igel et al. (2005) proposed a similar approach for the analysis of records observed by rotational seismometers. Subsequently, the seismic gradiometry technique was further developed for application to a twodimensional wave propagation problem (Langston 2007b) and to polarized waves (Langston and Liang 2008) and to improve its stability in the time domain (Langston 2007c). Compared with traditional array methods such as the semblance method (Neidell and Taner 1971), seismic gradiometry has the advantage that it estimates slowness as a spatially varying value, whereas other array methods usually assume a homogeneous plane wave incidence. In contrast to traditional array methods, which utilize phase differences for slowness estimation, seismic gradiometry models both the amplitude and phase of the observed wavefield. The possible separation of nonplane wave characteristics such as the radiation pattern and geometrical spreading is also a unique feature of the method.
In this paper, we first show that seismic gradiometry is a useful tool for reconstructing a spatially continuous seismic wavefield, although the method was originally applied (Liang and Langston 2009) to the estimation of the phase speed of surface waves at individual stations. We also propose that the divergence and rotation of the threecomponent seismic wavefield can be estimated by seismic gradiometry and used to characterize seismic wave propagation features. After synthetic tests using threedimensional numerical simulation results for an inhomogeneous structure model beneath Japan, we apply the proposed method to seismic waves in longperiod bands (>25 s) observed by Hinet stations.
Methods/Experimental
where N is the number of stations used for the estimation. Liang and Langston (2009) used a very similar observation equation to estimate Rayleigh waves for the case that the grid locations are collocated with seismic stations. Though their approach simplifies gradiometry Eq. (2), the estimation of the seismic wavefield is limited to the station location. In our approach, we use seismic gradiometry to reconstruct the seismic wavefield as a continuous 2D field irrespective of the station location. We note that Spudich et al. (1995) performed a similar series expansion to estimate surface strain and stress tensors for the 1992 Landers earthquake (M7.4) recorded by a dense array.
where K≡[k _{ u }, k _{∂x }, k _{∂y }]^{ T } is a part of the solution of Eq. (3). Equation (4) suggests that the displacement and its spatial gradients at a grid point are obtained from weighted averages of the observed waveform at nearby stations, where the weighting factor K is obtained from the station layout. We note that this is an overdetermined problem when three or more stations are used. Therefore, regularization by smoothing or damping is not necessary to solve this problem.
The quality of the estimation of the displacement and its spatial derivatives strongly depends on the locations of the stations relative to the grid point. The quality of the estimation improves as the number of stations used for Eq. (3) increases. On the other hand, the use of more distant stations may result in underestimation of the spatial derivatives of the waves, because we assumed a firstorder Taylor series expansion of the wavefield in Eq. (1) (Langston 2007a). We therefore used only those stations within 50 km, with a smoothly decreasing weighting factor.
Love and Rayleigh wave decomposition
The divergence and the vertical component of the rotation vector therefore consist of Rayleigh (PSV) and Love (SH) waves, respectively, whereas the horizontal components of the rotation are a mixture of Love and Rayleigh waves. By estimating these terms by seismic gradiometry, we can separate the Love and Rayleigh wave components from the observed ground velocity or displacement signal.
Slowness estimation
Notice that the spatial derivatives of the slowness vector are omitted in Eq. (10). In other words, we postulated that the slowness field changes slowly and that it can be assumed that the slowness is nearly constant at a grid point and the surrounding stations used for gradiometry estimation.
where u _{ t }, v _{ t }, and ∂_{ i } u _{ t } are vectors constituted by discrete time samples of ground displacement, ground velocity, and spatial gradients of ground displacement, respectively. We note that Liu and Holt (2015) used a very similar least squares method for estimating gradiometry parameters but with a damping constraint.
Because the estimation (12) is a pure timedomain method that uses a finite time window width, it does not require a longduration time window as do methods using a Fourier transform (Langston 2007b) or an analytic signal with a Hilbert transform (Langston 2007c). The estimation should be stable except for the case in which the denominator of Eq. (12), (u _{ t } ⋅ u _{ t })(v _{ t } ⋅ v _{ t }) − (u _{ t } ⋅ v _{ t })^{2}, is equal to or very close to zero. Such instability can occur only if the displacement time series u _{ t } is parallel to the ground velocity v _{ t }. However, such a situation does not arise if a single wave packet arrives at the array as assumed in Eq. (8), because the ground velocity phase of a packet always differs from that of the displacement by π/2. Thus, the gradiometry parameter estimation can be unstable only if multiple wave packets arrive at the array within the same time window. This drawback is intrinsic to the seismic gradiometry method itself, owing to the single wave packet assumption, not to the particular estimation method proposed here.
Results
Feasibility test with numerical simulation
For a feasibility test of wavefield estimation by seismic gradiometry, we first performed a synthetic test using the actual Hinet station layout. A threedimensional numerical simulation of seismic wave propagation was performed by using the finite difference method with the Japan Integrated Velocity Structure Model (JIVSM) of Koketsu et al. (2012) with a spatial extension to include an area not covered by the original model (Maeda et al. 2014). Technical details of the numerical simulation using the finite difference method are given by Maeda and Furumura (2013) and Maeda et al. (2013). We applied the seismic gradiometry method to synthetic seismograms at Hinet stations to estimate the spatial derivatives of the seismic wavefield, and also to estimate the divergence and rotation of elastic motion. The spatial derivatives of seismic waves in every grid of the numerical model were obtained during the numerical simulation by the finite difference method. Therefore, we compared the spatial derivatives between the numerical simulation and that estimated from the seismic gradiometry method.
Additional file 1. (MP4 13802 kb)
At every grid point, we estimated the spatial derivatives of the displacement by seismic gradiometry. We then calculated the divergence and rotation components of vector motion from these derivatives of the threecomponent seismograms. These divergence and rotation components were also calculated directly by finite difference simulation for comparison. In the estimation of divergence by seismic gradiometry, we used an approximation of Eq. (5) by assuming a Poisson solid (λ = μ) to mimic the realworld situation, in which the true inhomogeneous medium is unknown. In contrast, the exact divergence was calculated, including derivatives with respect to depth, in the case of the finite difference simulation in the same period band.
Additional file 2. (MP4 2476 kb)
Realworld application: 2005 OffTohoku outerrise earthquake
Next, we applied the same technique to real observations made during an earthquake that occurred in the east off Tohoku on 16 August 2005 (Mw 7.2). This earthquake generated peculiar surface waves that were observed in a wide area of Tohoku (Noguchi et al. 2016). We attempted to reconstruct the wavefield observed during this earthquake by applying seismic gradiometry to the Hinet records.
where u _{max} and v _{max} are the absolute values of the maximum amplitude of ground displacement and ground velocity traces in the whole time series, and ϵ ∼ 10^{− 6} is an empirically determined tolerance value. We estimated the gradiometry parameters when condition (13) was satisfied. This condition tended to not be met when the waves were not coherent at all, in particular, in the noise before the arrival of the first seismic wave. This result is consistent with the wavefield characterization in Eq. (8), for which it was assumed that a single wave packet dominated in the time window.
In realworld applications, it is also essential to avoid data from problematic stations. In particular, longperiod seismograms recorded by shortperiod seismometers with correction of their sensor responses are sometimes very noisy, even if the highfrequency components of the raw seismograms seem to be normal. To deal with this problem in an automated way, we introduced an iterative method to exclude outliers based on the rootmeansquared (RMS) amplitude of the noise level measured before the first arrival of the Pwave. First, the average and the standard deviations of this RMS amplitude were calculated, and the station having the largest standard deviation was excluded. In this procedure, only one station was excluded because a significant outlier may impact estimation of the average. Then, the average and standard deviations were calculated again using the dataset excluding the outlier. This procedure was repeated until all data were within three standard deviations of the average. This procedure is very similar in concept to the SmirnovGrubbs statistical test (Grubbs 1969), but we used an empirical threshold rather than the test statistic, because the RMS amplitude may not follow a normal distribution.
Additional file 3. (MP4 6558 kb)
At the elapsed time of t = 130 s (Fig. 8b), surface waves appeared near the epicenter in northeastern Japan. They were clearly recognized as a wave packet with wavelengths different from those of the body waves (marks D1 and E1). A corresponding difference in slowness was also seen, reflecting the difference between the apparent speed of the body waves and the phase speed of the surface waves (marks D2 and E2, respectively). At a late elapsed time (t = 230 s; Fig. 8c), the dispersion of the Rayleigh waves was clearly recognized by a gradual change of wavelength (mark F; longer in the SW to shorter in the NE) in the reconstructed wavefield. It is noteworthy that the major arrival directions of body waves in western Japan and surface waves in central Japan were not perfectly homogeneous but fluctuated in space, probably reflecting heterogeneity beneath Japan.
We found that the divergence and rotation were more sensitive to problematic station data. At the locations marked E1–E4, the seismogram was always oscillatory irrespective of the arrival of seismic waves (see Additional file 4: Movie S4). The amplitudes of the displacements and the derived wavefield (Fig. 8 and Additional file 3: Movie S3) did not show peculiar behaviors at these locations. Although this oscillatory feature is most probably due to problematic stations, it was not possible to automate their detection in the amplitudes observed before Pwave arrival.
Additional file 4. (MP4 3168 kb)
Discussion
Seismic gradiometry enables us to characterize the propagation direction by assuming that the wavefield consists of a single wave packet (Langston 2007b). Given this assumption, the method can successfully reproduce the major propagation direction and the slowness of direct waves and the early part of the surface wave coda. The estimated propagation direction of later coda was nearly random; thus, a more detailed investigation may be necessary to determine whether the assumption of a single wave packet is satisfied at later elapsed times. If the wavefield then consists of omnidirectional arrivals, suggesting a diffuse state (Weaver 1982), then wavefield characterization by seismic gradiometry would not be applicable, at least in its current form. Characterization of multiple incoming waves by this method should be addressed by future studies.
On the other hand, estimation of divergence and rotation by seismic gradiometry, newly proposed in this study, is not restricted by assumptions about wave behavior. Their estimation is always valid as long as the wavefield is continuous in space and its derivatives are correctly estimated. We showed that the spatiotemporal pattern of divergence and rotation, in particular the vertical component of the rotation vector, are extremely useful for inferring the wave constituents. Comparison of the reconstructed ground displacement or ground velocity wavefield with these divergence and rotation wavefields should be helpful for investigating the constituents of wave packets originated from inhomogeneous structures (e.g., Obara and Matsumura 2010; Domínguez et al. 2011; Noguchi et al. 2013, 2016; Maeda et al. 2014).
In the realworld application, we found that the recovery of a continuous, coherent wavefield was better achieved for the vertical component of the rotation vector than for the divergence (Fig. 9). This difference may be because, for divergence, the estimation of the derivative with respect to depth is indirect, whereas for the vertical component of the rotation vector, it is estimated directly from the derivatives in the horizontal directions. Recall that the derivative with respect to depth is estimated under the assumption of a tractionfree boundary condition, which is only approximately achieved at the Hinet stations installed at the bottom of boreholes (Obara et al. 2005); as a result, stable estimation may be distorted. If that is the case, this boundary condition becomes more valid for longer wavelengths in longperiod bands. This effect was apparently confirmed in the synthetic test (Fig. 6). The recovery of divergence was better in the longperiod band of 50–100 s than in the 25–50 s period band (Fig. 5).
This study focused on reconstructing the wavefield as a new method of analyzing seismic wavefields. If we concentrate on the arrival of direct waves, wavefield characterization by using gradiometry parameters leads directly to the estimation of phase speed and the azimuthal variation of surface waves from the great circle path (Liang and Langston 2009), which eventually leads to the estimation of the threedimensional inhomogeneous structure. Because the method provides a stable estimate of phase speed even from records of a single event, by compiling estimations by seismic gradiometry for many earthquakes having various back azimuth directions, it may be possible to analyze azimuthal anisotropy. Also, the use of the radiation pattern and geometrical spreading terms, which can be obtained from the gradiometry parameters, should be investigated in the future.
The estimation of the spatial derivative itself has room for further improvement. The first possible improvement is correction of site amplification. In highfrequency bands, the amplitudes observed at stations are significantly affected by local site amplification (Takemoto et al. 2012). Even at very low frequencies, where the surface wave mode dominates, the amplitude of the eigenfunction of surface waves is affected by local inhomogeneous structures near stations (Aki and Richards 2002). Although random fluctuation of the amplitude at stations can be minimized by least squares estimation of the spatial gradient, explicit correction of preestimated site amplification factors should allow more precise estimation of amplitude.
Because seismic gradiometry estimates slowness and other wave features from spatial gradients, its application might not be restricted to seismic waves in longperiod bands. In highfrequency seismograms with f > 1 Hz, the phase of the seismogram is usually very complicated owing to complicated wave scattering caused by smallscale inhomogeneity (Sato et al. 2012). Thus, applications of array methods to highfrequency seismograms are limited to records from local arrays of small spatial scale (e.g., Taira et al. 2007). However, seismogram envelopes are known to have similar shapes at nearby stations, and these characteristics have long been used for locating nonvolcanic deep lowfrequency tremors by the envelope correlation method (e.g., Obara 2002; Maeda and Obara 2009). We may therefore be able to apply the seismic gradiometry technique to seismogram envelope amplitudes in order to reconstruct the envelope wavefield and to estimate its propagation direction (see Appendix). The application of seismic gradiometry to seismogram envelope may be useful for investigating seismic wave energy flow in highfrequency bands.
The spatial modeling of waves could also be improved. We adopted a simple firstorder Taylor series in this study. Recently, Mizusako et al. (2014) showed that wave amplitude and its derivatives, including higher order derivatives, could be wellestimated by applying the Least Absolute Shrinkage and Selection Operator (LASSO) to the station layout of the Metropolitan Seismic Observation network (MeSOnet) (Kasahara et al. 2009). They showed that the continuous wavefield and its derivatives were quite wellestimated by their method, even though the MeSOnet station layout is much more irregular than the Hinet layout. Application of such cuttingedge mathematical technology will enable more accurate characterization of seismic waves, and, hence, the inhomogeneous structure beneath Japan.
Conclusions
In this paper, we attempted to visualize and characterize seismic wave traces at individual stations by using seismic gradiometry. The use of seismic gradiometry has three major benefits: (i) it enables us to treat seismic waves as a continuous 2D seismic wavefield rather than as a set of individual traces; (ii) the estimation of spatial derivatives enables us to separate divergence and rotation vector elastic oscillation; and (iii) it is useful for accurate estimation of slowness vectors as spatially varying field data.
Although wavefield visualization from seismic sensors has been done by applying spatial smoothing (Sheldrake et al. 2002; Maeda et al. 2014) or by using a set of individual traces (Trabant et al. 2012), seismic gradiometry makes it possible to estimate spatial gradients in horizontal directions as well, and these horizontal gradients are extremely useful for characterizing wave propagation. Also, the estimation process is linear and therefore can be done very quickly; thus, it is suitable for (semi) realtime seismic wavefield monitoring.
The results of our numerical experiments using finite difference simulations with a realistic 3D inhomogeneous medium model, together with the application of the method to realworld data, show that we can reconstruct a continuous and coherent wavefield with a period longer than 25 s by applying seismic gradiometry to Hinet records. Because it was previously confirmed that Hinet stations are capable of recording such longperiod data for farfield earthquakes of magnitude larger than 7 (Maeda et al. 2011), this method should be applicable to any such large earthquake if the records are not clipped by strong ground motion.
Notes
Abbreviations
 Hinet:

Highsensitivity seismograph network of Japan
 JIVSM:

Japan integrated velocity structure model
 MeSOnet:

Metropolitan Seismic Observation network
 NIED:

National Research Institute for Earth Science and Disaster Resilience, Japan
Declarations
Acknowledgements
We thank the data center of NIED for providing highquality seismic data from Hinet. The hypocenter location was obtained from the Japan Meteorological Agency hypocenter catalog. The Japan Integrated Velocity Structure Model (Koketsu et al. 2012) with modifications was used as the structural model for the numerical simulation. We used the gridded bathymetry dataset JTOPO30v2 provided by the Marine Information Research Center, Japan Hydrographic Association. The numerical simulation in this study was performed with the EIC computer system at the Earthquake Information Center of the Earthquake Research Institute, the University of Tokyo. We express our gratitude to Nori Nakata and an anonymous reviewer for their insightful comments and suggestions. This work was partly supported by the Earthquake Research Institute Cooperative Research Program (2015B01).
Funding
TM was funded by the Japan Society for the Promotion of Science (JSPS), KAKENHI Grant number15 K16306.
Authors’ contributions
TM, KN, and RT participated in the theoretical development, and TM carried out the data analysis and numerical simulation. KO conceived the study and contributed to the concepts and design of important intellectual components of the paper. All authors contributed to the drafting of the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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