UPC Barcelona
28 March 2003
Surface Wave Methods for Soil Characterisation
Sebastiano Foti Email:
[email protected] Web: www.polito.it/research/soilmech/foti
Politecnico di Torino
Contents • Rayleigh Waves - Introduction - Geometric Dispersion and Soil Characterization
• Standard Methods - Experimental Dispersion Curve - Inversion Process
• Stiffness and Damping - Attenuation of Rayleigh Waves - Transfer Function Method - Results at Leaning Tower of Pisa
• Final Remarks S. Foti
Surface Wave Methods for Site Characterisation
Contents èRayleigh Waves Introduction Geometric Dispersion and Soil Characterization
• Standard Methods Experimental Dispersion Curve Inversion Process
• Stiffness and Damping: Attenuation of Rayleigh Waves Transfer Function Method Results at Leaning Tower of Pisa S. Foti
Surface Wave Methods for Site Characterisation
Seismic Waves Body Waves
Surface Waves
Primary or Dilational Waves
Rayleigh Waves
Secondary or Shear Waves
Love Waves
Direction of Propagation S. Foti
Surface waves
(After Bolt, 1976)
Surface Wave Methods for Site Characterisation
Rayleigh Waves Particle Motion
Surface waves
(after Richart et al., 1970) S. Foti
Surface Wave Methods for Site Characterisation
Wave field generated by a point source acting on the surface of an elastic halfspace
Surface waves
% of total energy Rayleigh 67 Shear 26 Compression 7 Wave Type
(After Woods, 1968) S. Foti
Surface Wave Methods for Site Characterisation
Surface waves
Summary of main properties of R-waves Ø Easily generated and detected on the ground surface Ø 2/3 of the total energy released by a vertical harmonic point source acting on the surface of a homogeneous halfspace; Ø Reduced geometrical attenuation (1/√r) compared to other waves; Ø The propagation involves only a limited depth (~ 1 wavelength); Ø In homogenous linear elastic media: velocity of propagation is almost equal to V S and it is not frequency dependent Ø In vertically heterogeneous media: dispersive behaviour, i.e. phase velocity is function of frequency S. Foti
Surface Wave Methods for Site Characterisation
Surface waves and Characterization
Geometric Dispersion
?
Vertical particle motion Wavelength λ
VS1
Phase velocity VR
VS2> V S1
VR
VR = λ ⋅ f
VS3> V S2
Stiffness profile
Z
Frequency f
Z
Short wavelength High frequency
Dispersion Curve
Long wavelength Low frequency
Experimental
INVERSE PROBLEM
S. Foti
Surface Wave Methods for Site Characterisation
Surface waves and Characterization
Surface Wave Methods Seismograph or Signal Analyser Controlled or impact source
frequency components Low frequency vertical geophones
1
2
3
n
VS1 VS2 VS3
§ Testing depth ≈ 1/2 survey length § Resolution decreases at depth (problems in identifying thin layers) S. Foti
Surface Wave Methods for Site Characterisation
Rayleigh Waves and Characterization Field Testing Detection of motion on the ground surface
Signal Analysis
VR
Dispersion curve: Phase velocity of Rayleigh waves vs frequency ω
Inversion process
VS
G0
Variations of Shear Wave velocities with depth
G0 = ρ ⋅ V
2 S
Small Strain Stiffness profile (G0 vs depth) S. Foti
Z
Surface Wave Methods for Site Characterisation
Contents ü Rayleigh Waves Introduction Geometric Dispersion and Soil Characterization
èStandard Methods Experimental Dispersion Curve Inversion Process
• Stiffness and Damping: Attenuation of Rayleigh Waves Transfer Function Method Results at Leaning Tower of Pisa S. Foti
Surface Wave Methods for Site Characterisation
Rayleigh Waves and Characterization Field Testing Detection of motion on the ground surface
Signal Analysis
VR
Dispersion curve: Phase velocity of Rayleigh waves vs frequency
ω
Inversion process
VS
G0
Variations of Shear Wave velocities with depth
G0 = ρ ⋅ VS2 Small Strain Stiffness profile (G0 vs depth) S. Foti
Z
Surface Wave Methods for Site Characterisation
Experimental dispersion curve
The Steady State Rayleigh Method
(Jones, 1958)
λR is found moving the receiver (the receiver and the source are in phase)
For each frequency of operation of the source: S. Foti
VR = λ R ⋅ f
Surface Wave Methods for Site Characterisation
Experimental dispersion curve
The SASW (Spectral Analysis of Surface Waves) Method x 10
geophone 1 output
Signal Analyzer
1
1
0
0
-1
-1
-2
Far Receiver
geophone 2 output
x 10
Near Receiver
1
2
3
4
x 10 2
1
1
0
0
-1
-1
-2 1
2
time, s
3
4
5
-2 1.8
5
2
0
D
x 10 2
0
Impulsive, Sinusoidal or Random Noise Source
(a)
5
2
-2 1.8
(b)
1.82
1.84
1.86
1.88
1.9
1.82
1.84
1.86
1.88
1.9
5
time, s
Fast Fourier Transform Y1(ω)=FFT(y1(t)) Y2(ω)=FFT(y2(t))
X
Usually D=X Phase Velocity
Time Delay
Cross Power Spectrum
VR (ω) =X/ t(ω)
t(ω) = phase(Gy1y2 (f)) / ω
Gy1y2 = Y1(ω)* ⋅ Y2(ω)
Frequency range of acceptable data function of D (near field effects) S. Foti
Surface Wave Methods for Site Characterisation
SASW: rules for testing geometry
Experimental dispersion curve
Common source array
Common receiver midpoint array Heavy sources are used with larger spacing to obtain low frequency (long wavelength) information S. Foti
Surface Wave Methods for Site Characterisation
SASW: Assembling experimental data
Experimental dispersion curve
Averaging over frequency segments
D=12m D=6m 12m D D 3m D=3m D=30m D=18m 30m D== =6m 18m
Dispersion Dispersion curve curve 770000 dg ge e -- h ha am mm ss ll ee d me e rr (( D D= =3 3m m )) ha am mm me e rr (( D D= =6 6m m )) s ll ee dd gg ee -- h weight-drop(12m) weight-drop(12m)
velocity, m/s m/s phase phase velocity,
660000
weight-drop(18m) weight-drop(18m) weight-drop(30m)
550000
44 400 000 0
33 30000
220000
11 000 0 100
00 0
10 10
20 20 20
30 30 30
40 40
50 50
60 60 60
70 70
80 80 80
90 90
110000
frequency, frequency, Hz Hz S. Foti
Surface Wave Methods for Site Characterisation
Experimental dispersion curve
Analysis in the fk domain
Using a 2D Fourier transform data are taken in the fk domain. Such transform is often used in Geophysics to filter out ground roll components (mainly Rayleigh waves) that dominate the spectrum (from Doyle, 1995)
It can be shown that maxima in the spectra correspond to the dispersion curve
vR ( f ) =
2π ⋅ f k P=P
f = const
max
To ensure a sufficient resolution in the wavenumber domain a very high number of detection point would be necessary. This can be avoided adding zero-traces or using high resolution techniques for spectra estimation. S. Foti
Surface Wave Methods for Site Characterisation
Multistation Methods
Experimental dispersion curve
Impulsive or harmonic source Low frequency vertical geophones 2
D
X
2π ⋅ f vR ( f ) = k P=P
X
phase velocity, m/s
receiver offset (m)
2D FFT
each f
max
S. Foti
n
3
experimental dispersion curve
frequency,Hz Hz frequency,
1
time (s)
Seismograph or Signal Analyzer
experimental dispersion curve frequency, Hz
wavenumber,rad/m rad/m wavenumber, Surface Wave Methods for Site Characterisation
Analysis in the fk domain Mathematical proof (Tselentis & Delis, 1998)
Displacements by an impulsive point source (Aki & Richards, 1980) 1 s ( x, t ) = 2π
+∞
i (ωt − k S ( ω , x ) ⋅ e ∑ m ∫
m (ω ) x )
dω
−∞ m
discrete slant stack transform N
∑ s( xn ,τ + p ⋅ xn ) = 1 n=1 2π
=∑
1 = 2π
+∞
+∞
i (ωτ +ωpx −k S ( ω , x ) ⋅ e ∑ m n ∫ n
m (ω ) x n )
iωτ
N
i (ωp− km (ω ))⋅ xn
m n=1
Fourier transform over the time N −αm (ω)⋅ xn i( k − km (ω))⋅xn F (ω , k ) = ∑ Sm (ω ) ⋅ ∑ e ⋅e m n =1 S. Foti
I (ω )
instrument response
Pm (ω )
source spectrum
Rm (ω )
path response
1
Geometric attenuation
r
dω =
k m (ω ) =
Material attenuation
ω = ω ⋅ pm (ω ) VR m (ω )
−∞ m
∫ e ∑ ∑ Sm (ω , xn ) ⋅ e
−∞
e −αm ( ω) x S m (ω , x) = I (ω ) ⋅ Pm (ω ) ⋅ Rm (ω ) ⋅ x
αm
n=1
N
Experimental dispersion curve
dω
km
modal wavenumber
VRm
phase velocity
pm (ω )
slowness
For any given ω F=max per k=km
Surface Wave Methods for Site Characterisation
Experimental dispersion curve
Analysis in the fk domain Example: f=50.05 Hz k=0.9695 1/m VR=324.2 m/s 1.4
150 frequency=50Hz
1.2
particle velocity, m/s
frequency, Hz
1
100
50
50 Hz
0.8
0.6
0.4
0.9695 1/m 0.2
0
0 0.5
1
1.5
2
wavenumber, 1/m
S. Foti
2.5
3
0
0.5
1
1.5
2
2.5
wavenumber, 1/m
Surface Wave Methods for Site Characterisation
3
fk analysis results 1m
24
3
2
1
3m
3m
24
3
2
1
1m
1m
Experimental dispersion curve
3m
700
sledge-hammer (1m)
phase velocity, m/s
600
weight-drop (3m)
500
400
300
200
100 0
10
20
30
40
50
60
70
frequency, Hz S. Foti
Surface Wave Methods for Site Characterisation
fk vs. SASW dispersion curve
Experimental dispersion curve
700 SASW fk analysis
phase velocity, m/s
600
500
400
300
200
100 0
20
40
60
80
100
frequency, Hz S. Foti
Surface Wave Methods for Site Characterisation
Rayleigh Waves and Characterization Field Testing Detection of motion on the ground surface
Signal Analysis VR Dispersion curve: Phase velocity of Rayleigh waves vs frequency ω
Inversion process
VS
G0
Variations of Shear Wave velocities with depth
G0 = ρ ⋅ VS2 Small Strain Stiffness profile (G0 vs depth) S. Foti
Z
Surface Wave Methods for Site Characterisation
1 0.8
particle motion
Inversion process depth/wavelength
Approximate Inversion (SSRM) Mapping rule estimated experimental stiffness dispersion profile curve
λ*R
VR
1.1 V R*
0.5
0.6 0.4 0.2
S-wave R-wave
0
1
0.5
Poisson Ratio 1.5
Example with experimental data Vs (m/s) 100 200 300 400 500 600 700 0
VS
λ* R
5
3
10
Depth (m)
VR*
V/Vs
0
15 20 25
λR
Depth
30 35
S. Foti
Surface Wave Methods for Site Characterisation
Inversion process
The forward problem Soil model
Solution of the homogeneous eigenvalue problem (free Rayleigh modes) 460
H1 ρ1 G1 ν1 H3 ρ3 G3 ν3 ρ4 G4 ν4
Stack of linear elastic layers
phase velocity, m/s
H2 ρ2 G2 ν2
440 420 400 380 360 340 320 300 0
50
100
150
frequency, Hz
Considering an active source: mode superposition S. Foti
Surface Wave Methods for Site Characterisation
Inversion process
Usually νi and ρi are fixed and Hi and Gi (or Vsi) are the unknowns
Inversion Strategies Trial and error
Least Square
Global Search
Damped Neuronal Network Weighted Genetic Algorithms Occam's algorithm Simulated Annealing S. Foti
Surface Wave Methods for Site Characterisation
Inversion process
Inversion Codes SURF program (prof. Herrmann & co., S. Louis Un., USA) Forward problem: modified Haskell-Thomson algorithm for the homogeneous problem (no mode superposition) Inverse problem: Damped Least Square algorithm Matlab code by prof. Rix and Dr Lai Forward problem: modified R/T algorithm for the homogeneous problem (Hisada) + mode superposition Inverse problem: Occam’s algorithm
S. Foti
Surface Wave Methods for Site Characterisation
Inversion process
Shear wave velocity profile Shear Wave Velocity (m/s) 100
200
300
400
500
600
Damped Weighted Least Square Algorithm
700 800
5
Dispersion curve fitting 700 600
15
phase velocity, m/s
Depth (m)
10
20
25
500 400 300 200
30 100
CHT 35
S. Foti
0
10
20
30
40
50
60
frequency, Hz Surface Wave Methods for Site Characterisation
70
ENEA-Saluggia testing site Borehole CH Depth (m)
Material Description Sandy gravel
10
Borehole G
Sandy gravel
Depth (m)
Cover
Sandy gravel
Sand
10
40
Sand
20 30
Gravelly and/or silty sand
40 Clayey silt
50
Cover
Sandy gravel
20 30
Material Description
Gravelly and/or silty sand
50
Gravelly and/or silty sand Clayey silt
60 70 80 90 100
S. Foti
Silty clay Gravelly and/or silty sand Silty clay
Surface Wave Methods for Site Characterisation
SWM and P-wave refraction
Inverse stiffness profile 700
Vs [m/s]
500 0
400
200
400
600
800
1000
0
300 1
200 experimental numerical - modal velocities numerical - apparent velocity
100 0 0
100
200
300
400
500
600
700
frequency [Hz]
depth [m]
phase velocity [m/s]
600
2
3
4
Mode superposition è apparent dispersion curve 5
Seismic refraction cannot be applied in such stratigraphies S. Foti
Surface Wave Methods for Site Characterisation
Contents ü Rayleigh Waves Introduction Geometric Dispersion and Soil Characterization
ü Standard Methods Experimental Dispersion Curve Inversion Process
èStiffness and Damping: Attenuation of Rayleigh Waves Transfer Function Method Results at Leaning Tower of Pisa DR S. Foti
Surface Wave Methods for Site Characterisation
Energy dissipation and attenuation of seismic waves 1 ∆W (ω ) D (ω ) = 4π W (ω ) A( x ) = A0 ⋅ e
−
ωx 2VQ
= A0 ⋅ e −αx D=
S. Foti
αV ω
Surface Wave Methods for Site Characterisation
Rayleigh waves attenuation max particle velocity (mm/s)
10
experimental
8 6
geometric attenuation
4
1 r
2 0
0
10
20
30
40
50
60
distance (m)
Homogeneous medium
A1 ⋅ r1 ln( ) A2 ⋅ r2 α= r2 − r1 S. Foti
Definition of the non-geometric attenuation
Surface Wave Methods for Site Characterisation
Rayleigh waves attenuation The non-geometric attenuation of Rayleigh waves is closely linked to the dissipative characteristics of the medium
Heterogeneous medium: Attenuation Curve
Damping Ratio Profile DS
αR Inversion Process
ω S. Foti
Z Surface Wave Methods for Site Characterisation
Simultaneous measurement and inversion of dispersion and attenuation curves Field Testing Detection of motion on the ground surface
Signal Analysis
αR
VR
Dispersion and Attenuation of Rayleigh Waves ω
Inversion Process
ω DS
VS
Stiffness and Damping Ratio Profiles Z S. Foti
Z
Surface Wave Methods for Site Characterisation
Transfer Function or Frequency Response Function system
input
i
Linear and time invariant system
output
Frequency Domain U(ω)=H(ω)⋅I(ω)
u t
t
H(ω): all system information
Test configuration Signal Analyzer
Accelerometer Receiver
Shaker
Modelling the soil as a layered linear viscoelastic system, it is possible to operate a complex regression process on the experimental measurements of H(ω) to get VR e αR
r S. Foti
Surface Wave Methods for Site Characterisation
Transfer Function or Frequency Response Function system
input
Linear and time invariant system
output
i
Frequency Domain U(ω)=H(ω)⋅I(ω)
u t
t
H(ω) contains system information
Test configuration 1
r r
Fr (ω ) ~ F(r,ω ) = F1r (ω ) = F1 (ω ) S. Foti
It is possible to show that the experimental transfer function can be evaluated from the deconvolution in frequency domain of the seismic traces
Surface Wave Methods for Site Characterisation
Modelling the soil as a layered linear viscoelastic system
Displacements for a harmonic point source
w (r , ω ) = Fy ⋅ G(r ,ω ) ⋅ e i[ωt + Ψ (r ,ω )] Geometric spreading function G(r ,ω ) ≅
1 r
F y ⋅ e iω t (Lai & Rix, 1998)
Complex valued phase ω Ψ(r , ω ) ≅ K (ω ) ⋅ r = − iα R (ω ) ⋅ r VR (ω )
Analytical Transfer Function
w(r , ω ) ~ F(r , ω ) = = w(r1 , ω )
r1 r
⋅e
−i ⋅K (ω )⋅( r − r1 )
Regression of experimental data S. Foti
Surface Wave Methods for Site Characterisation
Regression process example Transfer Function @ 11.5 Hz 11
experimental regression -10
-20
0.6 0.4
00 10
20
30
40
50
60
receiver offset, m
K = 0.4876-i·0.0471 ω K (ω ) = − iα R (ω ) VR (ω ) S. Foti
experimental regression
0.8
0.2
frequency=11.5Hz -30
TF amplitude
TF phase, rad
0
10 10
20
30
40 40
50 50
60
receiver receiver offset, offset, m m
2π ⋅ f vR = = 148m / s Re(k ) α R = − Im(k ) = 0.0471 Surface Wave Methods for Site Characterisation
Experimental Curves and Inversion Attenuation Curve 0.14
attenuation, 1/m
180
experimental numerical
160
140
120 0
5
10
15
20
25
0.12 0.10 0.06 0.04 0.02 0
30
S. Foti
5
10
15
20
25
30
damping ratio
100 100 150 200 250 250 300 0
0
5
5
0
0.02 0.04 0.06 0.08
depth, m
10
20
0
frequency, Hz
shear wave Velocity, velocity, m/s
15
experimental numerical
0.08
frequency, Hz
depth, m
phase velocity, m/s
Dispersion Curve
10
SASW CHT
15 20
25
25
30
30
SASW Lab
Test site
Surface Wave Methods for Site Characterisation
Leaning Tower of Pisa site
S. Foti
Experimental results
Surface Wave Methods for Site Characterisation
Stratigraphy and results from Seismic CPT at the Leaning Tower of Pisa site
(After Jamiolkowski and Pepe, 2001) S. Foti
Surface Wave Methods for Site Characterisation
Transfer Function Method Results qc (MPa)
VS (m/s) 100 150 200 250 300
DR (%) 0
2
4
6
8
SASW CHT
SASW Lab
S. Foti
Surface Wave Methods for Site Characterisation
Final Remarks Ø Surface waves methods are cost and time effective and allow for accurate characterization Ø Non-invasive method (hard-to-sample soils) Ø Multistation methods show several advantages (near field effect, low frequencies, automation, stability, noise influence) with respect to SASW two-station approach Ø Multistation acquisition gives a chance to get an estimate of damping (Transfer Function Method) Ø Higher modes are important in some sites (the interpretation of the test becomes more complex) Ø Non uniqueness and resolution in SWM S. Foti
Surface Wave Methods for Site Characterisation
References Foti S. (2000) “Multistation Methods for Geotechnical Characaterization using Surface Waves “, PhD dissertation, Politecnico di Torino. (http://www2.polito.it/research/soilmech/sasw/SF_Phd_diss.pdf) Foti S. (2002) “Numerical and experimental comparison between 2-station and multistation methods for spectral analysis of surface waves”, RIG, n.1, 11-22 Foti S., Lancellotta R., Sambuelli L., Socco L.V. (2000) “Notes on fk analysis of surface waves”, Annali di Geofisica, vol. 43, n.6, 1199-1210 Foti S., Sambuelli L., Socco L.V., Strobbia C. (-) “Experiments of joint acquisition of seismic refraction and surface wave data”, submitted to the EEGS-ES Journal, under revision Gabriels P., Snieder R., Nolet G. (1987) “In situ measurements of shear-wave velocity in sediments with higher-mode Rayleigh waves”, Geophys. Prospect., vol. 35, , pp. 187-196 Lai C.G., Foti S., Godio A., Rix G.J., Sambuelli L., Socco L.V. (2000) “Caratterizzazione Geotecnica dei Terreni Mediante l'Uso di Tecniche Geofisiche”, RIG, Numero speciale: Sviluppi nell'esecuzione e nell'impiego delle indagini geotecniche, 99-118 Lai C.G., Foti S., Godio A., Sambuelli L., Socco L.V., Strobbia C. (2001) “Moderni metodi per l’esecuzione ed interpretazione delle indagini geofisiche” Relazione su invito, XVIII Conferenze di Geotecnica di Torino, Novembre 20-22, 50 pp Lai C.G., Rix G.J., Foti S., Roma V. (2002) “Simultaneous Measurement and Inversion of Surface Wave Dispersion and Attenuation Curves”, Soil Dynamics and Earthquake Engineering, Vol. 22, No. 9-12, pp. 923-930 Lai, C. G. and Rix, G. J. 1998. Simultaneous inversion of Rayleigh phase velocity and attenuation for near-surface site characterization. Technical Report GIT-CEE/GEO-98-2, Georgia Institute of Technology. (http://www.ce.gatech.edu/~grix/Lai_and_Rix_98.pdf) Nazarian S., Stokoe II K.H. (1984) “In situ shear wave velocities from spectral analysis of surface waves”, Proc. 8th Conf. on Earthquake Eng. - S.Francisco, vol. 3, Prentice-Hall, pp. 31-38 Stokoe K.H. II, Nazarian S., Rix G.J., Sanchez-Salinero I., Sheu J., Mok Y. (1988) “In situ seismic testing of hard-to-sample soils by surface wave method”, Earthq. Eng. and Soil dyn. II - Recent adv. in ground-motion eval. - Park City, vol. , ASCE, pp. 264-277 Stokoe K.H. II, Wright S.G., J.A. Bay, J.M. Roesset (1994) “Characterization of geotechnical sites by SASW method”, Geophysical Characterization of Sites (ISSMFE TC#10) by R.D. Woods, vol. , Oxford & IBH Publ., pp. 15-25 Strobbia C. (2002) “Surface Wave Methods in shallow geophysics”, PhD dissertation, Politecnico di Torino Tokimatsu K. (1995) "Geotechnical site characterisation using surface waves", Proc. 1st Int. Conf. on Earth. Geotechn. Eng., K. Ishihara, ed., Balkema, Rotterdam, pp. 1333-1368
S. Foti
Surface Wave Methods for Site Characterisation
