Forbriger, Thomas

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Low-frequency limit for H/V studies due to tiltTilt induced acceleration versus inertial accelerationThomas ForbrigerBFO (Schiltach) and GPI (Karlsruhe)October 20061 Low-frequency H/V peakThroughout the data recorded during the URS project(Ritter et al., 2005) in Bucharest city Ziehm (2006) ob-serves a low-frequency peak in H/V ratio near 0.2 Hz.The frequency of this peak is systematically varyingfrom lower frequencies in the north to higher frequen-cies in the south. This variation presumably is causedby the dipping of the interface between the cretaceousand the neogene with depth varying from 1000 m to2000 m (Maˆndrescu et al., 2004). Unfortunately in therecordings of most stations the H/V level strongly riseswith frequency decreasing below 0.15 Hz. This puts aneffective limit on the analysis of the peak at frequenciesbelow 0.2 Hz (Fig. 1). Since in some of the data thereappears to be a power-law relation between H/V ratioand frequency below 0.15 Hz, I discuss tilt as a possiblecause of this strong increase.Figure 1: H/V ratio calculated from the data of stationURS 14 obtained during the URS experiment in Bucharestcity. Time windows in the range from June 22nd 200421:00 UTC to June 23rd 2:00 UTC were used. Each graycurve displays the H/V ratio for one time window, while theblack curve gives the average and the dashed curves mark thestandard deviation. Two peaks are observed. One at 0.15 Hz,the other at 0.7 Hz. For frequencies below 0.15 Hz peaksmay be obscured by the increasing and unstable values ofH/V ratio. Courtesy of Julia Ziehm.2 Forces acting on an inertial sensorGravimeter, seismometers and tiltmeters all use a massas physical sensor. They sense inertial and gravita-tional acceleration and are principally unable to distin-guish between both sources of forces acting on them(Wielandt, 2002, secs. 2.1 and 3.3).Consider a displacement field ~u(~x, t). I distinguishthe vertical component uz(h,z, t) and the horizontal com-ponent uh(h,z, t), where z is the vertical coordinate (pos-itive in upward direction), h is the horizontal coordi-nate, and t is time. From this follows the inertial accel-eration in the horizontal directionah =∂2uh∂t2 (1)andaz =∂2uz∂t2 (2)in vertical direction. Assuming that the sensor sits onthe floor it additionally senses an acceleration due tothe component of gravity ~g = −geˆz (with eˆz pointingupward) that is coupled into its sensitive direction dueto ground tilt. For small tilt angles the contribution tothe vertical component ist negligible, while it can besignificant for the forces sensed by the horizontal com-ponents (Wielandt and Forbriger, 1999). For small tiltangles the resulting acceleration signal acting on thehorizontal component of the seismometer may be ap-proximated byaτ = g∂uz∂h . (3)If a˜z, a˜h, and a˜τ are the Fourier transforms of az,ah, and aτ, respectively, the Fourier transform of theseismometer’s vertical component output signal iss˜z(ω) = T (ω)a˜z(ω), (4)where T (ω) is the instrument’s complex response func-tion. Ands˜h(ω) = T (ω)(a˜h(ω)+ a˜τ(ω)), (5)is the Fourier transform of the seismometer’s horizontalcomponent output signal.3 Tilt contribution to H/V ratioH/V ratio is calculated from seismic recordings usuallybyAH/V(ω) =|R(ω)s˜h(ω)||R(ω)s˜z(ω)| =|a˜h(ω)+ a˜τ(ω)||a˜z(ω)| (6)and averaging over signals from several time windows.R(ω) is the response function of a deconvolution filterthat provides a displacement signal from the electricaloutput signal s(t). Hence the H/V ratio contains a tiltcontribution, which may be ignored at high frequenciessince the inertial effect rises with ω2. At low frequen-cies the tilt contribution is likely to dominate AH/V.12 Low-frequency limit for H/V studies due to tilt4 Point load on an elastic halfspaceThe elastic deformation of a homogeneous halfspacedue to a static point load is known as the”Boussinesquesolution“ in the theory of elasticity. According to Zu¨rn(2003) the solution to Boussinesque’s problem isuz(r,z) =− F4piµ1R(λ+2µλ+µ +z2R2)(7)for the vertical component andur(r,z) =− F4pi(λ+µ)1r(1+zR+λ+µµr2zR3)(8)for the horizontal component in cylindrical coordinateswith r =√x2+ y2 being the radial coordinate, R =√r2+ z2 being the spacial distance to the load, the load-ing force at the origin being~f =−Fδ(~x)eˆz (9)and λ and µ being Lame´s parameters for the homoge-neous halfspace at z < 0.For observations at the surface of the elastic halfs-paceuz(r,z = 0) =− F4piµ1rλ+2µλ+µ (10)for the vertical component andur(r,z = 0) =− F4pi(λ+µ)1r(11)for the horizontal component, which is valid for r > 0.If F(t) varies with time but so slowly that the de-formation instantaneously follows the variation of theload and elastic wave contribution to the displacementfield is negligible, this is the seismic near field. In thiscase the accelerations acting on the seismometer areaz(t,r,z = 0) =−¨F(t)rλ+2µ4piµ(λ+µ) , (12)ah(t,r,z = 0) =−¨F(t)r14pi(λ+µ) , (13)andaτ(t,r,z = 0) =g∂uz(t,r,z)∂r (14)= gF(t)r2λ+2µ4piµ(λ+µ) . (15)For the corresponding Fourier transforms of the seis-mometer’s output signals I obtains˜z(ω) =T (ω) ˜F(ω)14pi(λ+µ)ω2rλ+2µµ(16)ands˜r(ω) =T (ω) ˜F(ω)14pi(λ+µ)(ω2r+gr2λ+2µµ),(17)where ˜F(ω) is the Fourier transform of the point load’stime history F(t). This results inAHIV(ω) =ω2r+ gr2λ+2µµω2rλ+2µµ(18)=µλ+2µ +grω2. (19)The high-frequency limit islimω→∞ AHIV(ω) =µλ+2µ (20)and the low-frequency limit islimω→0AHIV(ω) =grω2(21)and rises with 1/ω2 for decreasing frequency ω. Bothcontributions are equal atωC =√grλ+2µµ. (22)Consider a point load at r = 10m distance, withλ = µ in the halfspace and g = 9.81ms−2. ThenωC = 2pi ·0.271s. (23)The H/V ratio will become AHIV > 2 at frequenciesf < 0.11Hz due to the tilt contribution, which is thenlikely to mask the ellipticity of surface waves. Addi-tional examples are displayed in Fig. 2.5 Propagating surface waveThe displacement field of the harmonic component ωof plane surface waves is given byuz(r, t) = A(ω) sin(k(ω)r−ωt) (24)andur(r, t) = A(ω)ε(ω) cos(k(ω)r−ωt), (25)where A(ω) is a frequency dependent amplitude, ε(ω)is the frequency dependent ellipticity, and k(ω) is thefrequency dependent wavenumber. For prograde wavesTilt induced acceleration versus inertial acceleration 30.11101000.1 1 10H/ V r at i ofrequency (Hz)H/V ratio for signals due to point load at distance r (λ=µ)r = 10 mr = 30 mr = 100 mr = 1 m1./3.9.81ms-2ω-2/(10.m)Figure 2: H/V ratio predicted by eq. (19) for a point loadat various distances r. Lame´’s constants are assumed to beequal (λ = µ).ε < 0 and ε > 0 for retrograde waves. For the accelera-tions acting on the seismometer I obtainaz(r, t) =−ω2 A(ω) sin(k(ω)r−ωt), (26)ar(r, t) =−ω2 A(ω)ε(ω) cos(k(ω)r−ωt), (27)andaτ(r, t) =gk(ω)A(ω) cos(k(ω)r−ωt). (28)For the H/V ratio this immediately results inAHIV(ω) =∣∣gk(ω)−ω2ε(ω)∣∣ω2(29)=∣∣∣∣ gωc(ω) − ε(ω)∣∣∣∣ , (30)where c(ω) = ω/k(ω) is the phase velocity.Assuming ε(ω) and c(ω) are bounded, the high-frequency limit provides the ellipticity.limω→∞ AHIV(ω) = ε(ω) (31)and the low-frequency limit islimω→0AHIV(ω) =gωc(ω). (32)Both cancel atωC =gεc(33)for retrograde waves. For prograde waves they are equalat ωC.Consider a surface wave with c = 2000ms−1, ε =0.67, and g = 9.81ms−2. ThenωC = 2pi ·1.2 ·10−3 1s. (34)Additional examples are displayed in Fig. 3.0.11100.001 0.01 0.1H/ V r at i ofrequency (Hz)H/V ratio for surface wave signalsε = 0.67, c = 2000 m/sε = 0.67, c = 1000 m/sε = 0.67, c = 500 m/sε = -0.67, c = 2000 m/sε = -0.67, c = 1000 m/sε = -0.67, c = 500 m/sFigure 3: H/V ratio predicted by eq. (30) for surface wavesof constant phase velocity c and constant ellipticity ε.6 ConclusionsGenerally we assume to observe the ellipticity of sur-face waves, when studying H/V ratio. Additionally tomicroseisms propagating as surface waves, we have toexpect point load varying with time (moving vehicles,buildings loaded by wind, etc.) close to the seismicinstruments in an urban environment. In this case wehave to expect a combination of effects described byeqs. (19) and (30) in the H/V ratio.It is however not correct to simply add the resultsfor AHIV caused be different sources. To estimate theH/V ratio individual contributions of surfaces waves,point loads and other sources must first be added inthe numerator and denominator of eq. (6) prior to tak-ing the ratio. The relative strength of the point loadcompared to the surface wave amplitude thus controlswhether the point load effect can take over the H/V ra-tio at low frequencies or not. This may explain thestrong scatter of H/V from different time windows atfrequencies below 0.1 Hz in Fig. 1. For the same rea-son we will never be able to observe the cancellation ofthe contributions to the horizontal component acceler-ation in eq. (29), neither as a hole in H/V nor as a peakin V/H. Horizontal acceleration due to other sources in-troduce a waterlevel in the total horizontal acceleration.Typical peaks in H/V ratio that are commonly at-tributed to Rayleigh wave ellipticity range between val-ues of 2 and 10. In the high-frequency range the peakscaused by Rayleigh wave ellipticity will dominate inany case. However, at low-frequencies tilt contribu-tion to the horizontal signal can dominate the H/V ratio.The effect for tilt due to surface waves may become sig-nificant at frequencies below 10 mHz, which is outside4 Low-frequency limit for H/V studies due to tiltthe bandwidth studied with the URS data. The effect oftilt induced by point loads at a distance of 10 m, how-ever, can dominate the H/V ratio and can explain theobservations made with URS data.7 AcknowledgementsI’m grateful to Julia Ziehm for providing Fig. 1 and toWalter Zu¨rn and Olivier Se`be for inspiring discussions.URS data was recorded with the KABBA arrays withinthe framework of CRC 461.ReferencesMaˆndrescu N., Radulian M. and Maˇrmureanu G., 2004.Site conditions and predominant period of seismicmotion in the Bucharest urban area. Rev. Roum.Ge´ophysique, 48: 37–48.Ritter J.R.R., Balan S., Bonjer K.P., Diehl T., For-briger T., Ma˘rmureanu G., Wenzel F. and WirthW., 2005. Broadband urban seismology in thebucharest metropolitan area. Seism. Res. Lett.,76(5): 574–580.Wielandt E., 2002. Seismic sensors and their calibra-tion. In: P. Bormann and E. Bergmann (editors),New Manual of Seismological ObservatoryPractice, GeoForschungsZentrum, Potsdam,Germany. <http://www.geophys.uni-stuttgart.de/seismometry/man html/index.html>, <http://www.geophys.uni-stuttgart.de/downloads/Postscript-files>.Wielandt E. and Forbriger T., 1999. Near-field seis-mic displacement and tilt associated with the ex-plosive activity of Stromboli. Annali di Geofisica,42(3): 407–416. <http://www-gpi.physik.uni-karlsruhe.de/pub/forbriger/>.Ziehm J., 2006. Seismisches Monitoring einer Großs-tadt, Analyse von H/V Spektralverha¨ltnissen.Diplomarbeit, Geophysikalisches Institut, Univer-sita¨t Karlsruhe. in preparation.Zu¨rn W., 2003. Boussinesque-problem: Point load onan elastic halfspace. A compilation of 12 pub-lished solutions to Boussinesque’s problem. Un-published manuscript.

Low-frequency limit for H/V studies due to tilt - Tilt induced acceleration versus inertial acceleration

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Low-frequency limit for H/V studies due to tilt - Tilt induced acceleration versus inertial acceleration

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