Post-processing of Continuous Shear Wave Signals April 28, 2011 Taeseo Ku Civil & Environmental Engineering EAS 4480

2 Introduction 1. Shear wave velocity (V s ) Second fastest wave & Directional and polarized Most fundamental wave to geotechnical engineering Depends on site-specific geostatic stress state 2. Applied data ‘ConeTec, Inc.’ produced continuous shear wave measurements with depth to 45m at Richmond, BC (Seismic CPT) Shear waves are collected at every 10cm vertical interval (Very short frequent interval !) Sampling time = 0.05 msec  f s = 20 kHz

Downhole Testing Oscilloscope Cased Borehole xx Test Depth Interval Horizontal Velocity Transducers (Geophone Receivers) packer Pump Horizontal Plank with normal load Shear Wave Velocity: V s =  R/  t z1z1 z2z2 tt R 1 2 = z x 2 R 2 2 = z x 2 x Hammer

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5 Post processing for continuous V s signals Signals are normalized by max. signal Signal process : de-trend and filter the raw signals (low/high cutoff frequency pass) A total of 418 data analyzed – bad signals are deleted Butterworth filter - [b,a] = butter(n, W n, 'type') Input ▪ n: order of filter  n = 4 ▪ W n : Cutoff frequency - use Nyquist frequency f N to normalize the input (f N = ½*f S = 10 kHz) ▪ W n =[w 1, w 2 ]/f N, band filter  [18Hz, 300Hz]/f N

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7 Auto-covariance Auto-covariance for signal at 45.3 meter depth x = [x 1, x 2, …, x n ] where x i is equally spaced in time

8 Power spectral density Power Spectral Density Estimate (signal at 45.3 meter depth) FFT fy = fft(y,nfft) f = 1/2*fs*linspace(0,1,n/2) Periodogram [Pxx,f] = periodogram(x,window,nfft,fs) LSSA : Lomb method [xp,xf] =lombscargle2(data,hifac,ofac) Observed Peak Frequency = 37.5 Hz

9 Post-processing for V s – Time domain Example : cross-correlation method in time domain Time shift Maximum covariance gives the time shift between two signals ! 45.2 & 45.3 meters Time shift sec

10 Post-processing for V s – Frequency domain Cross spectral analysis [Pxy,F]=cpsd(x,y,window,noverlap,nfft,fs) Signals at 45.2 & 45.3 meter depth Observed peak frequency = 37.5 Hz Phase = angle(Pxy)/(2*pi)*360; interp1(F, phase, 37.5) Lag: f = 37.5 Hz; T= sec ∆t = 8.924/360* = sec Signals at 45.2 & 45.3 m depth

11 Estimated V s profile Sensitive -Time lags are very small Running-mean Filter for ∆t Filter function: filtfilt y = filtfilt (b, a, x) describes filtering of vector x by y(n) = b(1)*x(n) + b(2)*x(n-1) b(n b +1)*x(n-n b ) -a(2)*y(n-1) a(n a +1)*y(n-n a ) Generate a n th order running mean filter coefficient vector  b=1/(n+1)*ones(1,n+1);

12 V s estimation using Running-mean filter 10 th order running mean filter 15 th order running mean filter

13 Relationship between V s and depth of overburden Empirical estimate from Lew & Campbell (1985, ASCE) Relationships are derived for various Quaternary age soils Over 270 V s surveys including refraction, downhole, and crosshole Regression analysis - V s =K(d+c) n d : depth K,c, and n : constants dependent on geotechnical classification Lew and Campbell,1985 Average V s

14 Regression analysis (Time domain : n = 10) 10 th order running mean filter Least-squares regression V s = *(Depth) (m/sec) Reduced major axis V s = 74.51*(Depth) (m/sec) Principal component V s = *(Depth) (m/sec) Y = a∙X c (Y = V s, X = depth) Transform to linear form  log(y) = c∙log(x) + log(a) Correlation coefficient: r = Coefficient of determination : R 2 = r 2 = 0.283

15 Regression analysis (Time domain : n = 15) 15 th order running mean filter Least-squares regression V s = *(Depth) (m/sec) Reduced major axis V s = 77.37*(Depth) (m/sec) Principal component V s = 97.82*(Depth) (m/sec) Y = a∙X c (Y = V s, X = depth) Transform to linear form  log(y) = c∙log(x) + log(a) Correlation coefficient: r = Coefficient of determination : R 2 = r 2 = 0.523

16 Regression analysis (Frequency domain : n = 10) 10 th order running mean filter Least-squares regression V s = 64.75*(Depth) (m/sec) Reduced major axis V s = 58.94*(Depth) (m/sec) Principal component V s = 63.12*(Depth) (m/sec) Y = a∙X c (Y = V s, X = depth) Transform to linear form  log(y) = c∙log(x) + log(a) Correlation coefficient: r = Coefficient of determination : R 2 = r 2 = 0.851

17 Regression analysis (Frequency domain : n = 15) 15 th order running mean filter Least-squares regression V s = 61.96*(Depth) (m/sec) Reduced major axis V s = 57.71*(Depth) (m/sec) Principal component V s = 60.71*(Depth) (m/sec) Y = a∙X c (Y = V s, X = depth) Transform to linear form  log(y) = c∙log(x) + log(a) Correlation coefficient: r = Coefficient of determination : R 2 = r 2 = 0.889

18 Summary Post processing for continuous V s signals ▪ Detrend data & band pass filtering ▪ Auto-covariance, PSD estimate V s evaluated by ‘cross-correlation’ & ‘cross-spectral analysis’ ▪ Time domain : find max. covariance between signals ▪ Frequency domain : cross spectral analysis – phase lags ▪ Sensitive results : apply n th order running-mean filter Regression analysis ▪ Transform to linear form (V s vs depth) ▪ LS regression, RMA regression, PC regression