Finite-difference Seismograms Research Articles

The finite-difference method in microseismic modeling: Fundamentals, implementation, and applications

The complexity of the seismic model around most reservoirs poses significant challenges for ray methods, even for 1D model structures. Scattering, waveguide, and near-field effects potentially create waveforms that can only be modeled adequately by the full elastic wave equation. Finite-difference modeling can form the basis of a new class of interpretation tool for these scenarios. Although used extensively in reverse time migration and full-waveform inversion algorithms, it is not commonly used in microseismic applications. Microseismic modeling places more stringent demands on the finite-difference method than its more conventional applications in surface seismic imaging. A finite-difference scheme can meet these requirements by carefully accounting for the main potential sources of error in modeled results. We examine the accuracy of finite-difference seismograms through comparisons with analytical results in a dip-slip modeling example. The value of this approach as an interpretation tool in microseismic modeling is demonstrated in a survey design example from the Fayetteville Shale, where scattering and waveguide phenomena have a significant effect on the observed amplitudes. We advise placing arrays at locations where amplitudes are robust to these effects when deploying real-time ray-based inversion methods in the field.

Hybrid Broadband Ground-Motion Simulation Using Scattering Green's Functions: Application to Large-Magnitude Events

We calculate near-source broadband (0-10 Hz) seismograms by combin- ing low-frequency three-dimensional (3D) finite-difference seismograms (0-0.5 Hz) computed in a 3D velocity model using site-specific scattering Green's functions for random, isotropic scattering media. The scattering Green's functions are convolved with a slip-rate function to form local scattering operators (scatterograms), which constitute the high-frequency scattered wave field. The low-frequency and high- frequency scatterograms are then combined in the frequency domain to generate broadband waveforms. Our broadband method extends the Mai et al. (2010) approach by incorporating dynamically consistent source-time functions and account- ing for finite-fault effects in the computation of the high-frequency waveforms. We used the proposed method to generate broadband ground motions at 44 sites located 5-100 km from the fault, for Mw 7.7 earthquake scenarios (TeraShake) on the south- ern San Andreas fault, which include north-to-south, south-to-north, and bilateral rupture propagation from kinematic and spontaneous dynamic rupture models. The broadband ground motions computed with the new method are validated by com- paring peak ground acceleration, peak ground velocity, and spectral acceleration with recently proposed ground-motion prediction equations (GMPEs). Our simulated ground motions are consistent with the median ground motions predicted by the GMPEs. In addition, we examine overturning probabilities for 18 precariously balanced rock sites (PBR). Our broadband synthetics for the Mw 7.7 TeraShake sce- narios show no preferred rupture direction on the southern San Andreas fault but are inconsistent with the existence of PBRs at several of the sites analyzed.

Effects of Irregular Structure of the Mississippi Embayment on Ground-Motion Amplification

The objective of this study is to evaluate effects of the presently available 3D velocity model of the Mississippi embayment structure on the amplification of seismic waves by using simulated finite-difference seismograms. Effects of both 2D and 3D embayment basement structures were considered. The 3D model included information of the near-surface velocities that were derived from the existing 1D velocity models. The 2D crustal model was taken from Catchings (1999), which extended from Saint Louis, Missouri, to Memphis, Tennessee. Finite-difference seismograms were simulated for point sources embedded at both ends of the 2D structure. These seismograms were examined to distinguish features like peak amplitude amplifications and duration of seismograms when the seismic waves propagated from the Mississippi Embayment toward the Illinois basin and vice versa. To establish a working 3D structure model of the embayment, we compiled geologic information of the region on material properties of the shallow structure and used the 3D model developed at the Center of Earthquake Research Institute (ceri), Memphis, as the starting model. The 3D model was used to generate finite-difference seismograms along several profiles for a M w 7.2 scenario earthquake occurring on the New Madrid fault zone. An equivalent 1D model, which included the basin materials, was also used to compare the 3D versus equivalent 1D ground motions simulated using the finite-difference method. To establish the amplification factors due to the surface sediments in the 3D model, finite-difference seismograms were also computed for a 1D hard-rock reference model. These 1D and 3D responses of the Mississippi embayment were used for estimating ground-motion amplification at sites where the depth to the basement is deeper than 500 m. Our investigation suggests that the deep structure of the Mississippi Embayment has little impact on long-period ground- motion amplitudes ( T ≥ 2 sec) for large earthquakes that rupture in the central part of the basin. This supports the hypothesis that for engineering purposes ground motions simulated based on the equivalent 1D crustal model are adequate for representing ground motions from future large earthquakes ( M w > 7) occurring in the New Madrid seismic zone.

Modelling of ground motion in the Higashinada (Kobe) area for an aftershock of the 1995 January 17 Hyogo-ken Nanbu, Japan, earthquake

We model the ground motion from an aftershock of the 1995 January 17 Hyogo-ken Nanbu (Kobe) earthquake to investigate basin edge effects on wave propagation in Higashinada ward, downtown Kobe. Point-source finite-difference seismograms calculated using a double-couple solution and a 2-D basin structure are compared with the ground-motion velocity seismograms recorded in a small station array deployed at sites within and outside the heavily damage zone in Higashinada ward. The comparison suggests that in the frequency range 0.1–2 Hz that was analysed, the observed spatial amplitude variation of the aftershock ground motion is attributable mainly to the basin edge effects. We found that the basin edge effect, caused by the superposition of the direct S wave and the basin-edge-diffracted waves, amplified the ground motion in a narrow zone that is offset by about 0.7 km from the basin edge.

Modeling strong motion in the Ashigara Valley for the 1990 Odawara, Japan, earthquake

Finite-difference seismograms are calculated for the 1990 Odawara earthquake. Using a basin structure derived from a geotechnical survey and a double-couple solution determined from the P first-motion data, a two-dimensional SH finite-difference calculation reproduces the observed spatial variation in amplitude and waveform of the transverse records at the Ashigara Valley Blind Prediction Sites for frequencies lower than 2 Hz. Comparison of the synthetics and the observed seismograms suggests that the observed amplitude variation of the transverse records in the Ashigara Valley is mainly due to the source radiation pattern rather than the basin structure. Several different double couples are used in the waveform modeling to show the sensitivity of ground motion to the source mechanism. We thus conclude that it is important to use an appropriate focal mechanism to model near-field ground motion.

Seismic-wave traveltimes in random media

SUMMARY A ray-theoretical relation is established between the autocorrelation function of the slowness fluctuations of a random medium and the autocorrelation function of the traveltime fluctuations on a profile perpendicular to the general propagation direction of an originally plane wave. Although this relation can be inverted exactly, it is preferable for applications to use the results of a forward calculation for a modified exponential autocorrelation function which represents slowness fluctuations with zero mean. The essential parameters of this autocorrelation function, standard deviation E and correlation distance a, follow by simple relations from the maximum and the zero crossing of the corresponding autocorrelation function of the traveltime fluctuations. The traveltime analysis of 2-D finite-difference seismograms shows that E and a can be reconstructed successfully, if the wavelength-tocorrelation-distance ratio is 0.5 or less. Otherwise, E is underestimated and a overestimated; however, both effects can be compensated for. The average traveltime, as determined from the finite-difference seismograms, is slightly, but systematically shorter than the traveltime according to the average slowness, i.e. the wave prefers fast paths through the medium. This is in agreement with results of Wielandt (1987) for a spherical low-velocity inclusion in a full-space and with results of Soviet authors, summarized by Petersen (1990). The velocity shift is proportional to E*, it has dispersion similar to the dispersion related to anelasticity, and it increases with the pathlength of the wave.

Lower crustal reflectivity patterns in wide‐angle seismic recordings

Many features of wide‐angle reflection data have been attributed to finely‐layered structure in the lower crust. Reflections with similar character, however, can be produced by small‐amplitude 2‐D random velocity variations. Finite‐difference seismograms for a lower crustal model with isotropic random velocity variations match field data recorded at the Black Forest, Germany. At source‐receiver offsets beyond about twice the target depth, waves scattered from a random zone produce the appearance of horizontal layering in wide‐angle data. Lower crustal reflectivity patterns can thus be explained by models that have lateral and vertical random fluctutations rather than fine‐scale layering.

Elastic finite-difference modeling of the 1971 San Fernando, California earthquake

Finite-difference seismograms calculated for the 1971 San Fernando earthquake show strong effects due to lateral variation in sediment thickness in the San Fernando valley and the Los Angeles basin. Using basin structure derived mostly from well logs and teleseismically determined source parameters, two-dimensional SH and P-SV finite-difference calculations reproduce the amplitude and duration of the strong-motion velocities recorded across the Los Angeles and San Fernando basins for the period range of 2 to 10 sec. The edges of basins nearest the seismic source show ground motion amplification up to a factor of 3 over the case without the basin and tend to convert direct shear waves into Love and Rayleigh waves that travel within the basins.

Finite-difference seismograms for SH waves

The accuracy of the finite-difference method for generating synthetic seismograms of SH wave propagation in cylindrically symmetric media is discussed. The finite-difference method has the advantage that arbitrary density and velocity fields in the medium may be specified. A point source is generated by a simple transformation of a line source. The accuracy of the finite-difference seismograms in flat- and dipping-layered media is confirmed by comparison with the generalized ray method. A source radiation pattern is inserted by introducing a “near-field” which has permanent displacement near the source. Strong motion synthetics are constructed with this new method for the 1968 Borrego Mountain earthquake as recorded at El Centro. Good fits to the data are achieved using the laterally varying model determined by a detailed refraction survey and the source parameters determined by teleseismic waveform modeling. Shallow faulting is no longer necessary to explain the long-period surface-wave development.

Finite difference seismograms for laterally varying marine models

Summary. The method of finite differences is applied to the elastic wave equation to generate synthetic seismograms for laterally varying seafloor structures. The results are compared with borehole seismic data from the Gulf of California (Deep Sea Drilling Project Site 485) in which lines were shot over flat and rough topography. The significant new phenomenon observed in both the synthetic seismograms and the field data is the generation of a ‘double head wave’ due to the interaction of the incident wavefront with the side of a hill and the flat seafoor adjacent to the hill. In these models the hills are on the order of a seismic wavelength in height and steep velocity gradients occur over distances comparable to wavelengths. Ray theoretical methods would not be suitable for studying such structures. True amplitude record sections are obtained by the finite difference method, which show for these models that the head wave generated at the flat seafloor adjacent to the hill is lower in amplitude than if the hill were not present and is lower in amplitude than the head wave generated at the hill. A second feature which is important for borehole receivers is the existence of the ‘direct wave root’ in the upper basement. This energy occurs below the sharp interface when the direct wave impinges on the interface from above. There is no corresponding Snell's law ray path for this energy and the energy is evanescent with depth in the lower medium. The properties of both the double head wave and the direct wave root are clearly demonstrated in the finite difference ‘snapshot’ displays.