An explanation of the large PGA value of 2013 M S7.0 Lushan earthquake at 51BXD station through topographic analysis

In M _S7.0 Lushan earthquake, a large amount of strong ground motion recordings were collected. In this paper, we analyze the recordings carefully. The abnormality of ground motion recordings is identified through a log linear regression. In the station of 51BXD, the PGA value has exceeded 1 g, which is the biggest peak ground acceleration (PGA) value obtained from all recordings in this earthquake. The log linear relation shows the PGA value in this station is abnormally large. As this station is located on the footage of a hill, the topographic amplification factor is explored in order to explain this abnormality. Through 3D numerical modeling using spectral element method with transmitting boundary conditions, the amplification factor is quantized. In this station, the topographic amplification is highly polarized in the direction of East–West which agrees with the empirical recordings. This research result suggests us in future directionality of topographic amplification should be considered in the aseismic design.     

Topographic versus stratigraphic amplification: mismatch between code provisions and observations during the L’Aquila (Italy, 2009) sequence

During the L’Aquila seismic sequence (Italy, 2009) we had the opportunity to install temporary accelerometric stations to study the role of seismic site amplification in damage enhancement. Two of the monitored sites, Castelnuovo and Navelli were also a good test for the recently introduced Italian seismic code (NTC08 2008) that prescribes an aggravation factor for slopes and ridges. Castelnuovo was an ideal situation to check the rule proposed for the distribution of amplification as a function of the position along a slope, while Navelli provided the possibility to test the almost equivalent factors that NTC08 sets for stratigraphic and topographic amplification (respectively up to 40 and 60 %). In neither case the observation matches code provisions. For Castelnuovo, there is a frequency dependence that shows as the code is over-conservative for short periods but fails to predict amplification in the intermediate range. For Navelli, the code provision is verified for long periods, but in the range around the site resonance frequency the stratigraphic amplification proves to be three times more important than the topographic one.     

Topographic site effects of Xishan Park ridge in Zigong city,Sichuan considering epicentral distance

Ground motion amplification induced by topography plays a vital role in engineering seismology. A topographic array of 8 accelerographs has been operating along the ridge in Xishan Park since 2007. The topographic site effects in Zigong city are studied based on the strong motion data of 2008 M_s 8.0 Wenchuan earthquake (the epicentral distance?=?225 km) and 2019 M_s 5.2 Zizhong earthquake (the epicentral distance?=?29 km). We compare the peak ground acceleration (PGA) of the two earthquakes and find that the PGA of Station 7#, which locates on a relatively steep slope, is amplified by 4.41 times comparing with the reference station in Zizhong earthquake, while this value is only 1.62 in Wenchuan earthquake. Fourier amplitude spectrum shows that the high frequency content of Zizhong earthquake is more abundant because of its smaller epicentral distance. By using the standard spectral ratio (SSR) method, we conclude that the amplification occurs because high-frequency ground motion is likely to resonate at small-scale features. Finally, the 3D numerical simulations are used to verify these conclusions. Our work indicates that more sophisticated numerical models need to be established for more accurate topographic site effects quantification. In addition, the influence of nearby topographic features should be considered when selecting reference stations.

     

Geophysical Images of the North Anatolian Fault Zone in the Erzincan Basin, Eastern Turkey, and their Tectonic Implications

The collision between the Arabian and Eurasian plates in eastern Turkey causes the Anatolian block to move westward. The North Anatolian Fault (NAF) is a major strike-slip fault that forms the northern boundary of the Anatolian block, and the Erzincan Basin is the largest sedimentary basin on the NAF. In the last century, two large earthquakes have ruptured the NAF within the Erzincan Basin and caused major damage (M _s = 8.0 in 1939 and M _s = 6.8 in 1992). The seismic hazard in Erzincan from future earthquakes on the NAF is significant because the unconsolidated sedimentary basin can amplify the ground motion during an earthquake. The amount of amplification depends on the thickness and geometry of the basin. Geophysical constraints can be used to image basin depth and predict the amount of seismic amplification. In this study, the basin geometry and fault zone structure were investigated using broadband magnetotelluric (MT) data collected on two profiles crossing the Erzincan Basin. A total of 24 broadband MT stations were acquired with 1–2 km spacing in 2005. Inversion of the MT data with 1D, 2D and 3D algorithms showed that the maximum thickness of the unconsolidated sediments is ~3 km in the Erzincan Basin. The MT resistivity models show that the northern flanks of the basin have a steeper dip than the southern flanks, and the basin deepens towards the east where it has a depth of 3.5 km. The MT models also show that the structure of the NAF may vary from east to west along the Erzincan Basin.     

Low-velocity zone and topography as a source of site amplification effect on Tarzana hill,California

Tarzana station is located in the foothills of the Santa Monica Mountains in California near the crest of a low (<20 m) natural hill with gentle slopes. The hill is about 500 m in length by 130 m in width and is formed of extremely weathered shale at the surface to fresh at depth. Average S-wave is about 250 m/s in the top 17–18 m, and S- and P-wave velocities significantly increase below this depth. According to the NEHRP classification based on V_S30∼300 m/s it is a site class D. Strong-motion instrumentation at Tarzana consisted of an accelerograph at the top of the hill, a downhole instrument at 60 m depth, and an accelerograph at the base of the hill. More than 20 earthquakes were recorded by at least three instruments at Tarzana from 1998 till 2003. Comparisons of recordings and Fourier spectra indicate strong directional resonance in a direction perpendicular to the strike of the hill. The dominant peaks in ground motion amplification on the top of the hill relative to the base are at frequencies ∼3.6 and 8–9 Hz for the horizontal components. Our hypothesis is that the hill acts like a wave trap. This results in an amplification at predominant frequencies f=V/4 h (h is layer's thickness) at f∼3.6 Hz for S-waves (using average V_S17=246 m/s and h=17 m) and f∼7.9 Hz for P-waves (using average V_P17=535 m/s and h=17 m). As was shown by Bouchon and Barker [Seismic response of a hill: the example of Tarzana, California. Bull Seism Soc Am 1996;86(1A):66–72], topography of this hill amplifies and polarizes ground motion in the frequency range of 3–5 Hz. Hill acts as a magnifying polarizing glass: It polarizes ground motion in the direction perpendicular to the strike of the hill and also amplifies ground motions that had been also amplified by a low-velocity layer.     

Amplification of earthquake ground motion by steep topographic irregularities

The problem of amplification of seismic waves by surface topographic irregularities is addressed through analytical and numerical investigations. First, a closed‐form expression for estimating the fundamental vibration frequency of homogeneous triangular mountains is obtained, using Rayleigh's method. Subsequently, numerical modelling based on the spectral element approximation is used to study the 3D seismic response of several real steep topographic irregularities excited by vertically propagating plane S‐waves. A topographic amplification factor is obtained for each case, by a suitable average of the ratio of acceleration response spectra of output vs input motion. The 3D amplification factors are then compared with those derived by 2D analyses as well as with the topographic factors recommended in Eurocode 8 for seismic design. Copyright © 2002 John Wiley & Sons, Ltd.     

3D Response analysis of an instrumented hill at Matsuzaki, Japan, by a spectral method

Strong ground motion observed at an instrumented hill site is first analysed through the standard (SSR) and the horizontal-to-vertical (HVSR) spectral ratio techniques. A reasonable agreement is found between these approaches. The observations are then compared with 3D numerical simulations, performed with a highly efficient numerical code based on a spectral method, that allowed for reasonable computer times also on a PC. The observed amplification is significantly higher than that computed with a 3D homogeneous model of the mountain, suggesting that local response is governed by large-scale and small-scale soil heterogeneities rather than by topographic site effects. The introduction of a local near-surface inclusion of nonhomogeneous soil material under one of the recording stations has not significantly improved the numerical results. The observed data are also compared with the results of simplified simulations, either using 2D homogeneous models or coupling the 3D response with a 1D local soil profile. The results of such simplified approaches are discussed and their usefulness is emphasised.     

Experimental study of the ground motion on a large scale topographic hill at Kitherion (Greece)

We present the results of an experiment about the effects on the seismic ground motion of a hill of large dimensions (6km long, 3km large and 700m high) near Corinth (Greece). We installed 7 seismometers across the hill, and analyzed the ground motion with different methods: the classical spectral ratios (CSR) and the horizontal to vertical spectral ratios calculated both on noise (HVNR) and earthquake data (RF). The amplification at the resonant frequency (around 0.7Hz) is low, as expected, but one of the two stations installed at the top exhibits large amplification (up to a factor of 10) at 3Hz. The comparison between the H/V results and the CSR ones suggests that the former method is able to point out the fundamental frequencies of a hill. Amplifications and spatial localization predicted by numerical 3D modeling are consistent with the experimental data only at frequencies lower than 1Hz.     

Directional dependence of site effects observed near a basin edge at Aegion,Greece

We study site effects using 520 weak motion earthquake records from a vertical array in Aegion, Greece. The array is inside a basin, has four stations in soil, and one in bedrock (178 m depth). The site is marked by high seismicity and complex surface geology. We first use the records to establish the downhole accelerometer orientations and their evolution with time. Then we estimate site effects using empirical spectral ratios with and without a reference site (standard and horizontal-to-vertical spectral ratio). We find significant site amplification which cannot be accounted for by 1D model predictions, along with a significant difference in the amplification level between the two horizontal components. These are indications of 2D effects, namely surface waves generated at the basin edge. The difference in amplification between the horizontal components is maximised when these are rotated with respect to the orientation of the basin edge. The strongest amplification takes place in the direction parallel to the basin edge (SH, or out-of-plane motion), and is up to 2 times higher than in the perpendicular direction (SV, or in-plane motion). This directional effect on the amplification is corroborated by numerical 2D modelling using incident SH and SV waves, with the former possibly generating strong Love waves. In the records, the directionality is clear for windows containing the largest amplitudes of the records (S waves and strong surface waves), while it tends to vanish for coda-wave windows. This directionality is also observed when using response spectral ratios rather than Fourier ratios. We compute soil-to-rock amplification factors for peak ground acceleration (PGA) and find it is significantly higher than what is predicted by current design codes. We attribute this difference to the basin edge amplification, linear soil behaviour, and to the inability of simple scalar values like PGA to describe complex amplification effects. Finally, we analyse the earthquake records at a surface station near the slope crest and do not observe significant topographic amplification.     

Slope topography effects on ground motion in the presence of deep soil layers

An extensive investigation has been made into the interaction between topographic amplification and soil layer amplification of seismic ground motion. This interaction is suggested in the literature as a possible cause for the differences between topographic amplification magnitudes observed in field studies and those obtained from numerical analysis. To investigate this issue a numerical finite element (FE) parametric study was performed for a slope in a homogeneous linear elastic soil layer over rigid bedrock subjected to vertically propagating in-plane shear waves (Sv waves). Analyses were carried out using two types of artificial time history as input excitation, one mimicking the build-up and decay of shaking in the time histories of real earthquake events, and the other to investigate the steady-state response. The study identified topographic effects as seen in previous numerical studies such as modification of the free-field horizontal motion, generation of parasitic vertical motion, zones of alternating amplification and de-amplification on the ground surface, and dependence of topographic amplification on the frequency of the input motion. For the considered cases, topographic amplification and soil layer amplification effects were found to interact, suggesting that in order to accurately predict topographic effects, the two effects should not be always handled separately.     

Exploration of S-wave velocity profiles at strong motion stations in Eskisehir,Turkey, using microtremor phase velocity and S-wave amplification

We have explored 1D S-wave velocity profiles of shallow and deep soil layers over a basement at strong motion stations in Eskisehir Province, Turkey. Microtremor array explorations were conducted at eight strong motion stations in the area to know shallow 1D S-wave velocity models. Rayleigh wave phase velocity at a frequency range from 3 to 30 Hz was estimated with the spatial autocorrelation analysis of array records of vertical microtremors at each station. Individual phase velocity was inverted to a shallow S-wave velocity profile. Low-velocity layers were identified at the stations in the basin. Site amplification factors from S-wave parts of earthquake records that had been estimated at the strong motion stations by Yamanaka et al. (2017) were inverted to the S-wave velocities and Q-values of the sedimentary layers. The depths to the basement with an S-wave velocity of 2.2 km/s are about 1 km in the central part of the basin, while the basement becomes shallow as 0.3 km in the marginal part of the basin. We finally discussed the effects of the shallow and deep sedimentary layers on the 1D S-wave amplification characteristics using the revealed profiles. It is found that the shallow soil layers have no significant effects in the amplification at a frequency range lower than 3 Hz in the area.     

Site response studies and seismic microzoning in the Middle Aterno valley (L��aquila, Central Italy)

Following the April 6th, 2009 Abruzzo mainshock, the Italian Civil Protection Department promoted a multidisciplinary study aimed at developing seismic microzonation maps for post-earthquake reconstruction planning. In the framework of this project, a Working Group, including the authors, was assembled to carry out a microzonation study on six villages located in the Middle Aterno valley. This paper focuses on the villages of Castelnuovo and Poggio Picenze, which experienced MCS intensity values of IX–X and VIII–IX, respectively. 1D and 2D linear equivalent site response analyses were carried out on representative geological cross-sections through the damaged centres and the expansion zones. The subsoil models resulting from geological, geotechnical and geophysical investigations were calibrated by comparing numerical amplification functions, in the linear range, with horizontal-to-vertical spectral ratio derived from both aftershocks and noise recordings. The input motions adopted for the analyses were five artificial accelerograms compatible with three response spectra obtained from the Italian seismic code, as well as from ad hoc probabilistic and deterministic studies. The results were expressed in the form of horizontal profiles of amplification factors in terms of peak ground acceleration, F_PGA, as well as of the Housner intensity, FH, in two different range of periods; this latter parameter was shown to be almost independent of the input motion and allowed to express the dependency of site amplification on the frequency range. The amplification factors computed along the representative geological sections were finally extended with a rational procedure to the surrounding areas to draw Grade-3 microzonation maps.     

Numerical study on basin-edge effects in the seismic response of the Gubbio valley,Central Italy

The Gubbio basin in Central Italy is a intermountain basin of extensional tectonic origin, typical of Central and Southern Apennines, characterized by moderate seismicity. The strongest recorded event within the area is a magnitude 5.7 earthquake which occurred on 29 April 1984 along the Gubbio fault, bordering the eastern side of the basin. The main objective of this study is to analyze the features of earthquake ground motion as related to basin-edge effects, by performing physics-based numerical simulations of the 1984 earthquake through a high-performance spectral element code. The simulated ground motions are found in reasonable agreement with the recorded motions when using the kinematic source model developed by Ameri et al. (Bull Seismol Soc Am 99:647–663, 2009), with a rise-time equal to 1 s and a nucleation point located in the middle of the fault. Pronounced differences were noted between records from the basin and adjacent sites at outcropping bedrock, owing to both the strong impedance contrast between soft alluvial sites and bedrock formations (lithostratigraphic amplification), as well as lateral discontinuities related to the 2D/3D geometry of the basin (generation of surface waves). Since the fault was located beneath the basin, 1D amplification effects were found to be more relevant than those associated with the generation of surface waves from the basin edge. Finally, an envelope delay spectrum was computed for the simulated ground motions, showing that surface waves are excited in the frequency band of 0.2–0.8 Hz with a significant increase of ground motion duration within the basin.     

1920年海原地震滑坡密集区的地震动场地效应研究

地震动作为引起地震灾害的原动力,常常通过造成建筑物倒塌、山体滑坡等形式引起大量人员伤亡和财产损失。1920年海原8½级地震,在震中距80 km远的西吉—静宁交界的黄土丘陵区引发了大量的山体滑坡,并造成重大人员伤亡和财产损失。在分析海原地震高烈度区滑坡分布特征的基础上,通过场地调查和数值计算等方法,研究典型滑坡密集场地的地质条件及地震反应特征。研究表明起伏地形和黄土厚度不均等因素造成丘陵山体两侧地震反应的差异,从而导致地震滑坡在斜坡土体较厚的一侧成群连片发育。海原地震造成的滑坡密集区的地形地貌、岩土性质、土层结构等条件决定了该地区地震动随局部场地条件变化非常迅速,地层场地效应和地形场地效应联合作用加剧了斜坡地表的地震动放大作用,增加了触发地震滑坡的动力。     

Earthquake ground motion modeling of induced seismicity in the Groningen gas field

A physics‐based numerical approach is used to characterize earthquake ground motion due to induced seismicity in the Groningen gas field and to improve empirical ground motion models for seismic hazard and risk assessment. To this end, a large‐scale (20 km × 20 km) heterogeneous 3D seismic wave propagation model for the Groningen area is constructed, based on the significant bulk of available geological, geophysical, geotechnical, and seismological data. Results of physics‐based numerical simulations are validated against the ground motion recordings of the January 8, 2018, M_L 3.4 Zeerijp earthquake. Taking advantage of suitable models of slip time functions at the seismic source and of the detailed geophysical model, the numerical simulations are found to reproduce accurately the observed features of ground motions at epicentral distances less than 10 km, in a broad frequency range, up to about 8 Hz. A sensitivity analysis is also addressed to discuss the impact of 3D underground geological features, the stochastic variability of seismic velocities and the frequency dependence of the quality factor. Amongst others, results point out some key features related to 3D seismic wave propagation, such as the magnitude and distance dependence of site amplification functions, that may be relevant to the improvement of the empirical models for earthquake ground motion prediction.     

Attenuation of P- and S-waves in the Chamoli Region, Himalaya, India

The attenuation properties of the crust in the Chamoli region of Himalaya have been examined by estimating the frequency-dependent relationships of quality factors for P waves (Q_α) and for S waves (Q_β) in the frequency range 1.5–24 Hz. The extended coda normalization method has been applied on the waveforms of 25 aftershocks of the 1999 Chamoli earthquake (M 6.4) recorded at five stations. The average value of Q_α is found to be varied from 68 at 1.5 Hz to 588 at 24 Hz while it varies from 126 at 1.5 Hz to 868 at 24 Hz for Q_β. The estimated frequency-dependent relations for quality factors are Q_α = (44 ± 1)f^(0.82±.04) and Q_β = (87 ± 3)f^(0.71±.03). The rate of increase of Q(f) for P and S waves in the Chamoli region is comparable with the other regions of the world. The ratio Q_β/Q_α is greater than one in the region which along with the frequency dependence of quality factors indicates that scattering is an important factor contributing to the attenuation of body waves in the region. A comparison of attenuation relation for S wave estimated here (Q_β = 87f^0.71) with that of coda waves (Q_c = 30f^1.21) obtained by Mandal et al. (2001) for the same region shows that Q_c > Q_β for higher frequencies (>8 Hz) in the region. This indicates a possible high frequency coda enrichment which suggests that the scattering attenuation significantly influences the attenuation of S waves at frequencies >8 Hz. This observation may be further investigated using multiple scattering models. The attenuation relations for quality factors obtained here may be used for the estimation of source parameters and near-source simulation of earthquake ground motion of the earthquakes, which in turn are required for the assessment of seismic hazard in the region.     

Simulation of Ground Motion from Strong Earthquakes of Kamchatka Region (1992–1993) at Rock and Soil Sites

To estimate the parameters of ground motion in future strong earthquakes, characteristics of radiation and propagation of seismic waves in the Kamchatka region were studied. Regional parameters of radiation and propagation of seismic waves were estimated by comparing simulations of earthquake records with data recorded by stations of the Kamchatka Strong Motion Network. Acceleration time histories of strong earthquakes (M _w = 6.8–7.5, depths 45–55 km) that occurred near the eastern coast of Kamchatka in 1992–1993 were simulated at rock and soil stations located at epicentral distances of 67–195 km. In these calculations, the source spectra and the estimates of frequency-dependent attenuation and geometrical spreading obtained earlier for Kamchatka were used. The local seismic-wave amplification was estimated based on shallow geophysical site investigations and deep crustal seismic explorations, and parameters defining the shapes of the waveforms, the duration, etc. were selected, showing the best-fit to the observations. The estimated parameters of radiation and propagation of seismic waves describe all the studied earthquakes well. Based on the waveforms of the acceleration time histories, models of slip distribution over the fault planes were constructed for the studied earthquakes. Station PET can be considered as a reference rock station having the minimum site effects. The intensity of ground motion at the other studied stations was higher than at PET due to the soil response or other effects, primarily topographic ones. At soil stations INS, AER, and DCH the parameters of soil profiles (homogeneous pyroclastic deposits) were estimated, and nonlinear models of their behavior in the strong motion were constructed. The obtained parameters of radiation and propagation of seismic waves and models of soil behavior can be used for forecasting ground motion in future strong earthquakes in Kamchatka.     

Estimation of site effects using strong motion data of BYTNet array in Turkey

Simultaneous estimation of effects of source, propagation path, and local site amplification was carried out using observed strong motion records in a frequency range from 0.8 to 20 Hz for the purpose of empirical evaluation of the local site effects in different geological conditions in the northwestern part of Turkey. The analyzed data are S-wave portions of 162 accelerograms from 39 shallow events observed at 14 sites of BYTNet array. A spectral separation method was applied to the observed S-wave spectra. The solutions for source spectra, inelasticity factor of propagation path for S-waves (Q _s-value), and factor of site amplification at each site were obtained in a least squares sense. In the analysis, we assumed that the factor of the site amplification at a reference site is the same as that of theoretical amplification of S-waves to the soil model whose bottom layer has an S-wave velocity around 2.15 km/s. The estimated Q _s-value of the propagation path is modeled as Q _s(f)?=?87.4f^0.78. The estimated site amplifications are characterized into three groups. The sites in the first group belong to rock site with no dominant peaks at a frequency range of 2 to 10 Hz. The second group of hard soil sites is characterized with moderately dominant peaks at a frequency of 5 Hz. The last group for soft soil sites has common peaks at a frequency of 4 Hz with larger amplitudes than those in the hard soil group. We, then, compare the amplifications with average S-wave velocity in top 30 m of the shallow S-wave profiles and proposed linear empirical formula between them at each frequency. We, furthermore, inverted the observed amplification factors into S-wave velocity and Q _s-value profiles of the deep soil over the basement.     

Numerical Modeling and Observations of Tsunami Waves in Alberni Inlet and Barkley Sound, British Columbia

Alberni Inlet is a long and narrow fjord adjacent to Barkley Sound on the Pacific Coast of Vancouver Island, Canada. Port Alberni, at the head of the inlet, was affected in 1964 by the largest Pacific tsunami waves in Canadian history. We use observations and results from two numerical models to investigate the resonant characteristics of the region and amplification of tsunami waves in Barkley Sound and Alberni Inlet. The first model (A) was forced at its open boundary with a stationary autoregressive signal, similar to the observed background noise. The second model (B) used an initial sea-level deformation from a potential earthquake off California in the southern segment of the Cascadia Subduction Zone, producing transient tsunami waves. Spectral, cross-spectral and frequency-time (f-t) analyses of the observations were used to examine the resonant properties and topographic response of the local area. The respective results show large admittance functions over a wide 0.5–0.9 cph frequency band, implying a low Q factor but high amplification of arriving waves. This unusual behavior is a result of two effects: A quarter-wave resonance of the system for its fundamental Helmholtz mode and amplification due to the narrowing of the channel cross section from Barkley Sound to Alberni Inlet. The model A numerical results agree favorably with the observations, indicating an energetic resonant mode at frequency of ~0.53 cph (112 min), with its nodal line located near the entrance to Barkley Sound and amplification factor value close to 12. The results from the tsunami propagation model (B) yield spectral characteristics similar to those from the model A and from the observations. The maximum tsunami current speed for this scenario is 2.4 ms^?1 in Sproat Narrows, which divides Alberni Inlet into two parts, while the largest computed wave height is 1.6 m in the northern Alberni Inlet, in the area of Port Alberni.     

Comparison of 3D, 2D and 1D numerical approaches to predict long period earthquake ground motion in the Gubbio plain, Central Italy

In this work we studied the performance of different numerical approaches to simulate the large amplifications of long period earthquake ground motion within the Gubbio plain, a closed-shape intra-mountain alluvial basin of extensional tectonic origin in Central Italy, observed during the Umbria-Marche 1997 seismic sequence. Particularly, referring to the Sep 26 1997 M_w6.0 mainshock, we considered the following numerical approximations: (a) 3D model, including a kinematic model of the extended seismic source, a layered crustal structure, and the basin itself with a simplified homogeneous velocity profile; (b) 2D model of a longitudinal and transversal cross-section of the basin, subject to vertical and oblique incidence of plane waves with time dependence at bedrock obtained by the 3D simulations; (c) 1D model. 3D and 2D numerical simulations were carried out using the spectral element code GeoELSE, exploiting in 3D its implementation in parallel computer architectures. 3D numerical simulations were successful to predict the observed large amplification of ground motion at periods beyond about 1 s, due to the prominent onset of surface waves originated at the southern edge of the basin and propagating northwards. More specifically, the difference of 3D vs 2D results is remarkable, since the latter ones fail to approach such large amplification levels, even when an oblique incidence of plane waves is considered.