Comparison between two methods for forward calculation of ambient noise H/V spectral ratios Antonio García-Jerez(1), Francisco Luzón(1), Francisco J. Sánchez-Sesma(2), Dario Albarello(3), Enrico Lunedei(3), Michel Campillo(4), Miguel A. Santoyo(5), Úrsula Iturrarán-Viveros(6) 1
Departamento de Física Aplicada, Universidad de Almería, 04120 Almería, Spain, E-mail:
[email protected],
[email protected]
2
Instituto de Ingeniería, Universidad Nacional Autónoma de México, CU, Coyoacán 04510 D.F., Mexico, E-mail:
[email protected]
3
Dipartimento di Scienze della Terra, Università di Siena, Via Laterina 8, 53100 Siena, Italy, E-mail:
[email protected],
[email protected] Université J. Fourier, BP 53, 38041 Grenoble Cedex, France, E-mail:
[email protected]
5 Departamento
de Ciencias, Universidad Nacional Autónoma de México; C.U., Coyoacán 04510 D.F., Mexico.
SUMMARY
1. THE H/V SPECTRAL RATIO AS DESCRIBED BY DSL
Horizontal-to-vertical (H/V) spectral ratio of ambient vibrations is a valuable tool for seismic prospecting for situations in which both a dense spatial sampling and a lowcost procedure are required. Unfortunately, the method still lacks of a unanimously accepted theoretical basis and different approaches are currently being used for inversion of the ground structure from the measured H/V curves.
P1 ( ) P2 ( )
2
xmin
P3 ( )
The main peak frequency matches the first vertical-SH resonance apart from slight variations (see Fig. 1 left for DFA). Nevertheless, the connection between peak amplitude and impedance contrast seems to be non-trivial (Fig. 1 right). Moreover, if DSL were the best theoretical model, peak amplitudes would show high dependence on the source positions (Fig. 4). Some significant differences appear, in the surface-wave approximation, at frequencies higher than the fundamental S-wave resonance: the [H/V]DSL curves show sharper peaks and troughs (associated with higher modes) in comparison with their [H/V]DFA counterparts (Fig. 2). Theoretical considerations (not shown) demonstrate that contributions of higher surface-wave modes cannot be neglected even for the DFA. Moreover, we have checked that powers of i) the fundamental Rayleigh mode in H and V components ii) the sum of higher Rayleigh modes in H and V components iii) the sum of all Love modes, are all of them O(f) as f →∞ for the DFA (a single layer over halfspace with 1 > 2 was assumed and the dependence of ES k -3 on the frequency was disregarded).
Model with several significant impedance contrasts The experiments performed with model M3 suggest that DFA and DSL method lead to closer results when near sources are removed (from the DSL computations) and surface waves play the major role. Nevertheless, further investigation on these aspects is still necessary.
2 1
11
(0; x1 ,0; ) G22 (0; x1 ,0; )
G
2 2
31
2
2
(0; x1 ,0; ) 2 G33 (0; x1 ,0; ) 2
2 3
2 3
2
G13 (0; x1 ,0; )
2
x dx 1
1
H/V spectral ratio (Sánchez-Sesma et al., 2011)
x dx 1
1
[ H / V ]( )
2 1
2 3
A
Rm / k Rm
P1 P2 m 1 / 2 m
2 1
2 3
A
/ 2 1
2
A
Lm
/ k Lm
2
h (m) 25 5000 ∞ h (m) 5 25 50 ∞
VS (m/s) 200 1000 2000 VS (m/s) 200 228-1520 2000 VS (m/s) 30 100 150 500
0.01-0.49 0.333 0.257 M3 VP (m/s) 500 500 500 1500
QS ∞ ∞
f
/f
HV
DFApeak 0SH
1.9 2.5 2.5 (g/cm3) 1.9 2.5 2.5 / 4 1 1 1 1
QP 500 500 500 QP 500 500 500 QP 500 500 500 500
QS 500 500 500 QS 500 500 500 QS 500 500 500 500
Figure 2. Solid line: [H/V]DFA(f/f0SH) for surface waves. Dashed line: [H/V]DSL(f/f0SH) following Arai & Tokimatsu (2004). A load relationship 12= 22= 32/2 has been used in this work for all DSL computations. Models M1* were considered in these calculations. Note that DSL often generates sharper curves.
(f
) / [TF
DFA DFApeak
0.55 0.5
k ES ARm , ALm
m
2
2j
Parametric study for DFA and DSL (M2* models)
(f
(m=3 stands for vertical component) Shear wavenumber
Average energy of shear waves Medium response for Rayleigh and Love waves (Harkrider, 1964) Rayleigh wave ellipticity (as a real number) Total surface variance density of random sources (Lunedei & Albarello, 2010) j- component relative surface variance density
Near source effects in DSL method (Model M2) xmin= 0 m
)/2]
SH 0SH
0.55
7
0.5
1.02
0.45
(g/cm3) 0.333 0.333 0.257 M2*
QP ∞ ∞
2 ES k Im[Gmm (x A ; x A ; )] 3
um (x, ) m-th component of displacement at x
Figure 1. Main peak characteristics for DFA (M1* models and full wavefield simulations)
0.45
1
0.4
6
0.4 0.98
0.35
1/ 2
h (m) 25 5000 ∞
0.00-0.45 0.3449 M2
/ 2 0.7391 1
2 Im[G11 (x; x; )] Im[G33 (x; x; )]
1 P1 P2 Im[G11 (x; x; )] m ARm m2 ALm 4
3. SOME SYNTHETIC TESTS
M1* VS / VS2 0.05-0.50 1
P1 ( ) P2 ( ) P3 ( )
[ H / V ]( )
Models used in the numerical experiments
4h/VS1 (s) 1 ∞
Pm (x A , ) um (x A , ) u (x A , ) * m
1 P3 Im[G33 (x; x; )] m ARm 2
H/V spectral ratio
2 / k Rm Rm m
2 3
P1 P2 P3
Power spectral density in m component
SURFACE WAVE PART (elastic model)
2
0.96
0.3 4
0.25
0.94
0.2
0.92
0.2
0.15
0.9
0.15
0.1
0.88
0.05 0
0.1
0.2
0.3
0.4
0.25 3
Figure 4. Variation of H/V by the DSL model when sources surrounding the station are removed up to xmin=10 m radius.
2
0.1 0.05 0
0.5
xmin=10 m
5
0.35
0.3
1 / 2 = 0.45
In spite of the different underlying hypotheses, the DFA and the DSL method provide similar H/V curves for models with a dominant impedance contrast (M1*, M2, M2*, Figs. 2 to 4). Nevertheless peak amplitudes may differ (Fig. 3). Surface waves are the dominant contribution for frequencies larger than the fundamental SH resonance one.
2
G
2
1/ 2
2 m
Mathematics of the DFA are significantly simpler than the DSL method formulation. Multiple evaluation of Bessel functions is avoided in the DFA.
Models with a dominant impedance constrast
2 2
SIMPLIFIED SURFACE WAVE PART (Arai & Tokimatsu, 2004) (elastic model, far field terms only, 1 = 2 , …)
Overall aspects
The DSL method generalizes some closely related techniques (e.g. Field and Jacob, 1993; Arai and Tokimatsu, 2004). It accounts for both surface and body wave contributions and for some cases of anisotropic illumination ( 1 ≠ 2 ). In this latter situation, a simple formula for the horizontal-component power spectrum is preserved provided that two orthogonal power spectral densities are averaged.
2 1
xmin
The DSL method should be better suited in situations in which ambient vibrations are generated by surface sources and effects of multiple scattering are negligible. On the contrary, the DFA should be more appropriate whenever multiple scattering plays a major role.
Power spectral density in each Cartesian component (Lunedei & Albarello, 2010)
P3 m 1 /
4. DISCUSSION AND CONCLUSIONS
ui (x A , )u (xB , ) 2ES k Im Gij (x A ; xB ; ) 3
* j
1 / 2 = 0.35
The main difference between these methods is the way in which the amount of energy injected in each type of wave mode is established. In the DFA method, that ratio is prescribed by the energy equipartition principle, whereas it is controlled by the statistical source (surface point loads) characteristics in the DSL method.
Cross-correlation and imaginary part of Green function are proportional
Diffuse Field Approach
1 / 2 = 0.25
Two major approaches for interpretation of H/V spectral ratios in a layered medium are discussed in this work. The first one consists of a description of the wavefield as generated by Distributed Surface Loads (DSL method; Lunedei & Albarello, 2010). The second possibility is the Diffuse Field Approach (DFA, Sánchez-Sesma et al., 2011) in which ambient noise is considered as a diffuse wavefield, taking advantage of the proportionality between its Fourier-transformed autocorrelation (power spectrum) and the imaginary part of the Green function, when source and receiver are the same.
2. THE H/V SPECTRAL RATIO AS DESCRIBED BY DFA
Randomly distributed surface point loads
1 / 2 = 0.15
6 Facultad
de Geofísica y Meteorología, Facultad de Ciencias Físicas, Ciudad Universitaria, UCM, 28040 Madrid, Spain.
1 / 2 = 0.05
4 LGIT,
S23A-2230
0.1
0.2
1
0.3
0.4
0.5
1
1
10
0
10
0
Figure 3. Top and central panels show full-wavefield DFA and DSL spectral ratios for variations in two M2 parameters (M2* set). We used xmin=0. The surface-wave version of the DSL method is shown in the lower panels.
1
Tests for a model with several impedance contrasts
1
10
10
10
xmin= 0 m
0
xmin=5 m
10
1
10
0
10
1
10
0
10
1
2 1 = 0
3
4
1
2 1 = 0.1
3
4
1
2 1 = 0.2
3
4
1
2 1 = 0.3
3
4
1
2 1 = 0.4
3
4
Figure 5. Same as Fig. 4 for model M3 in which two important impedance contrasts are found at 5 m and 80 m depth. If near sources are removed, both the DFA and the DSL results are similar (double peak in the H/V). Otherwise the models agree for frequencies higher than 1 Hz only.
5. REFERENCES Albarello D. & Lunedei E. (2011). Structure of an ambient vibration wavefield in the frequency range of engineering interest ([0.5, 20] Hz): insights from numerical modelling, Near Surface Geophysics, 9, in press, on-line on http://nsg.eage.org/earlyonline.php, doi:10.3997/1873-0604.2011017 Arai H. & Tokimatsu K. (2004). S-wave velocity profiling by inversion of microtremor H/V spectrum, Bull. Seismol. Soc. Am. 94, 5363. Field E. & Jacob K. (1993). The theoretical response of sedimentary layers to ambient noise, Geophys. Res. Lett. 20, 2925-2928.
Harkrider D. G. (1964). Surface waves in multilayered elastic media. Part 1, Bull. Seismol. Soc. Am. 54, 627-679. Lunedei E. & Albarello D. (2010). Theoretical HVSR curves from full wavefield modeling of ambient vibrations in a weakly dissipative layered Earth, Geophys. J. Int. 181, 1093–1108. Sánchez-Sesma F. J., Rodríguez M., Iturrarán-Viveros U., Luzón F., Campillo M., Margerin L., García-Jerez A., Suarez M., Santoyo M. A. & Rodríguez-Castellanos A. (2011). A theory for microtremor H/V spectral ratio: application for a layered medium, Geophys. J. Int. 186, 221-225.
