Accounting for non-stationary frequency content in Earthquake Engineering: Can wavelet analysis be useful after all?

Academic excellence for business and the professions Accounting for non-stationary frequency content in Earthquake Engineering: Can wavelet analysis be useful after all? Agathoklis Giaralis Senior Lecturer Dept. of Civil Engineering City University London, London, UK agathoklis@city.ac.uk Alessandro Margnelli PhD Candidate Dept. of Civil Engineering City University London, London, UK Associate Director at AKTII, London, UK alessandro.margnelli@akt-uk.com 4 nd Risk, Hazard, Uncertainty, etc. Workshop 3-4 June 016, Hydra, Greece

Introduction / Motivation Typical earthquake accelerograms exhibit a timeevolving frequency composition due to the dispersion of the propagating seismic waves, and a timedecaying intensity after a short initial period of development. 6 th ~7 th second: 1 st ~ nd second: 7 zero crossings 15 zero crossings

Introduction / Motivation Transient signals encountered in earthquake engineering and structural dynamics are inherently non-stationary: Both their frequency content and amplitude vary with time. Earthquake induced strong ground motion (accelerograms) GMs: Exhibit a time-evolving frequency composition due to the dispersion of the propagating seismic waves, and a time-decaying intensity after a short initial period of development. Response time histories of yielding structures under seismic excitation: Their evolving frequency content carries information about the possible level of (global) structural damage (e.g. degradation of the effective natural frequencies). Such signals call for a joint time- frequency analysis; for it is clear that their time- dependent frequency content cannot be adequately represented by the ordinary Fourier analysis. 3

Introduction / Motivation The ordinary Fourier Transform (FT) provides only the average spectral decomposition of a signal.

Introduction / Motivation Time-frequency analysis tools provide meaningful non-stationary signal representations 5

Presentation Outline Introduction / Motivation The Continuous Wavelet Transform (CWT) The wavelet-based mean instantaneous period (MIP) MIP of Recorded Seismic ground motions (GMs) MIP of Hysteretic Response Signals The alpha α angle of the average MIP The α as a GM property for the evolving frequency content Concluding remarks

The continuous wavelet transform (CWT) The continuous wavelet transform (CWT) given by the equation 1 W a, b f t * t b dt 1 a * t b a W x a, b 1/ x t dt a a decomposes any finite energy signal f(t) onto a basis of functions a : scaling generated parameter by scaling a single mother wavelet function ψ(t) by the b : localization scale parameter α and by shifting it in time by the parameter b. : analyzing or mother wavelet ab, 1 t b a a Variable size windows are employed Long duration windows capture lower frequencies (large scales) Short duration windows are used to capture higher frequencies (small scales) Heisenberg s uncertainty principle holds Scale Time

The continuous wavelet transform (CWT) s( t) s( t) Such an analysis results in a three-dimensional spectrum having the wavelet w coefficients plotted versus time and scale (scalogram). A certain wavelet-dependent s( w) dwrelationship between scale and frequency should be established to yield a wavelet- based spectrogram. dt dt t be arbitrarily high. w S( w) dw Uncertainty Principle Fourier Pairs tance a frequency t component ˆis located. We can e present in which time intervals. 1 t b Fourier Pairs ˆ exp i b Reciprocal relationship between scale-frequency: Frequency= constant/scale Scale Time

The continuous wavelet transform (CWT) Is CWT useful? 3x 1999 Chi Chi, Taiwan (station TCU098)

The continuous wavelet transform (CWT) Is CWT useful?

Period (s) The continuous wavelet transform (CWT) 0 0.0039065 0.0039065 But we need to know what we are aiming for: Time or Frequency (resolution/ smoothness/bias)??? Harmonic Wavelets Morlet Wavelets Global Wavelet Spectrum Global Wavelet Spectrum HW MW FFT FFT 0.007815 0.007815 0.01565 0.01565 0.0315 0.0315 Period [s] 0.065 0.065 0.15 0.15 1 0.5 0.5 0.8 0.5 0.5 0.5 0.6 Input Signal 0.4 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [s] 0. [s] 0 Time (s) 1 0 0.5 1 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 Power [(m/s ) 1 ] Power [(m/s ) ] 0 Power -0. -0.4-0.6-0.8-1 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

The continuous wavelet transform (CWT) But we need to know what we are aiming for: frequency Time or Frequency (resolution/smoothness/bias)??? Uncertainty principle Resolution trade-off time Wavelet shape Smoothness etc.

The continuous wavelet transform (CWT) Modified complex Morlet wavelets At scale α and time position b the modified Morlet wavelet is given by M t b 1 t b c exp i t b a a a a b b Its Fourier transform is a shifted Gaussian function, that is: ˆ M b exp b a a a c ia b 4 The central (pseudo-) frequency observed at scale α is usually computed by c o a The constant Ω b controls the bandwidth of the Gaussian function in the frequency domain

The continuous wavelet transform (CWT) Modified complex Morlet wavelets The scaling operation s( t) dt by α<1 moves the central w S( w) dw frequency Ω c /α w t) towards dt higher frequency s( w) dw levels. be arbitrarily It also compresses high. (narrows) the time domain waveforms which nce a frequency component is located. We can leads to reduced resolution in the present in which time intervals. frequency domain (uncertainty principle).

The continuous wavelet transform (CWT) Generalized harmonic wavelets A generalized harmonic wavelet of (m,n) scale and k position in time is constructed as a box-like function in the frequency domain (Newland, 1994), that is: T o i kt o m n ˆ exp, m, n, k n m n m To To 0, otherwise ; where T o is the effective duration of the signal to be analyzed. In the time domain it is a complex- valued function given by t k sin n m To n m t k ( m, n), k t exp i m n t k To n m n m To n m Central frequency at scale (m,n): (m+n)π/t o Bandwidth in the frequency domain at scale (m,n): (n-m)π/t o

The continuous wavelet transform (CWT) Generalized harmonic wavelets Harmonic wavelets of different scales can have arbitrarily chosen bandwidths throughout the frequency domain. This is because the scales are defined by two parameters (m,n), as opposed to one (α) in the case of common wavelets used in the context of the CWT.

Morlet: Constant Q The continuous wavelet transform (CWT)

The wavelet-based mean instantaneous period (MIP) Can we make CWT more useful? -> GM non-stationary frequency content characterization? Mean Period (e.g. Rathje et al.1998) is defined starting from DFT as: A wavelet based time-varying instantaneous period (MIP) can be defined as (Margnelli/Giaralis 015): Evolution in time of the Mean period Frequency range: [0.5 5]Hz

0.01565 Period (s) The wavelet-based mean instantaneous period (MIP) 0.0 0.0315 Morlet Wavelet Pow er Spectrum - Tm=0.89044 - TmGWS=0.8133 - TmIMP=1.3186 - TmIMP-Max=1.145 - TmIMP-Min=0.53713 - Action = LOMA-PRIETA-HOLLISTER-19-HDA165 Global Wavelet Spectrum MW FFT 0.065 0.15 Mean Instantaneous Period (MIP) Period [s] 0.5 0.5 1 Box that delimits the results between 5% to 95% of Husid function in time and 0.5Hz<f<5Hz in frequency 4 8 3 Power [(m/s ) ] 16 3 0. 0.1 0 Tm=0.89s Time (s) 0 5 30 35 Scale-averaged w avelet pow er - Husid t5%=6.115 t95%=16.395 0 0 5 30 35 40 0 0.5 1 Power Normalised Power [(m /s 3 )] Averaging in scale (scale-averaged wavelet power) Averaging in time (global wavelet spectrum)

The wavelet-based mean instantaneous period (MIP) MIP is a generalization of T m : Temporal averaging of MIP should yield T m Morlet Wavelet Pow er Spectrum - Tm=0.36113 - TmGWS=0.31005 - TmIMP=0.3359 - TmIMP-Max=0.55931 - TmIMP-Min=0.16803 - Action = IMPERIAL-VALLEY-PLASTER-18-H-PLS045 Global Wavelet Spectrum 0.01565 0.0315 MW FFT 0.065 0.15 Period [s] 0.5 0.5 1 4 8 Power [(m/s ) ] 16 0.06 0.04 0.0 0 4 6 8 10 1 14 16 18 Scale-averaged w avelet pow er - Husid t5%=4.81 t95%=14.445 0 0 4 6 8 10 1 14 16 18 0 0.5 1 Normalised Power [(m /s 3 )]

The wavelet-based mean instantaneous period (MIP) MIP is a generalization of T m : Temporal averaging of MIP should yield T m Harmonic Wavelet Morlet Wavelet Pow er Pow Spectrum er Spectrum - Tm=0.36113 - Tm=0.36113 - Tmw - H-IMP=0.36684 TmGWS=0.31005 - Tmw - TmIMP=0.3359 H-GWS=0.35973 - TmIMP-Max=0.55931 - Tmw H-IMP-Max=0.4653 - TmIMP-Min=0.16803 - Tmw H-IMP-Min=0.8481 - Action = IMPERIAL-VALLEY-PLASTER-18-H-PLS045 - Action = IMPERIAL-VALLEY-PLASTER-18-H-PLS045 Global Global Wavelet Wavelet Spectrum Spectrum Period [s] Period [s] 0.01565 0.01565 0.0315 0.0315 0.065 0.065 0.15 0.15 0.5 0.5 0.5 0.51 MW FFT HW FFT 1 4 Power [(m/s ) ] 8 4 16 8 0 4 6 8 10 1 14 16 18 0 4 6 8 10 1 14 16 18 Scale-averaged w avelet pow er - Husid t5%=4.81 t95%=14.445 0.06 0.04 0.0 0 0 4 6 8 10 1 14 16 18 0 0.5 1 Normalised 0 Power 0.5[(m /s 3 )] 1 Normalised Power [(m /s 3 )]

The wavelet-based mean instantaneous period (MIP) MIP is a generalization of T m : Temporal averaging of MIP should yield T m

The wavelet-based mean instantaneous period (MIP) MIP may not correspond to any actual frequency component in multi-chromatic signals it only coincides with the wavelet ridge for mono-chromatic signals 3

The wavelet-based mean instantaneous period (MIP) How useful MIP is? It does capture what we expect to see AND it is only a time-history rather than a matrix (CWT)

ω= 50 rad/s or T=0.15s 50 Wavelet analysis of hysteretic response signals How useful CWT is for hysteretic structural response? Moving average period elongation Frequency law - Sw =0.009411 45 40 k(t) [rad/s] 35 30 5 0 t [s] t=0s t=15s ω= 15 rad/s or T=0.4s

ω= 50 rad/s or T=0.15s 50 Wavelet analysis of hysteretic response signals How useful CWT is for hysteretic structural response? Moving average period elongation 0.6 0.4 Frequency law - Sw =0.009411 Artificial Accelerogram - Sw=0.009411 45 0. 40 ddug 0 k(t) [rad/s] 35 30-0. -0.4-0.6 5-0.8 t [s] 0 t [s] t=0s t=0s t=15s ω= 15 rad/s or T=0.4s t=15s

Period [s] A [m/s] 10 5 0 Acceleration [m/s] fy = 0.7 T = 0.5 S0 = 1 =0 Sw =0 N /s T input= 0.16 T input1= 0 Wavelet analysis of hysteretic response signals How -5 useful CWT is for studying the hysteretic structural response? -10 Moving average period elongation 0.0315 0.065 0.15 0.5 0.5 1 4 8 Elasto-plastic SDOF oscillator with fundamental pre-yielding period: T=0.5s Wavelet Power Spectrum Global Wavelet Spectru f y =0.7 T=0.15s T=0.4s T=0.5s Yield(red) - SAWP(blue) 0 t=0s Yield Ny=6 Ay=0.16 sec Iy=0.96 nsec - 0 1000 t=15s Power [(m/s ) ]

A [m/s] 10 5 0 Acceleration [m/s] fy = 0.5 T = 0.5 S0 = 1 =0 Sw =0.009411 N /s T input= 0 T input1= 0 Wavelet analysis of hysteretic response signals How -5 useful CWT is for studying the hysteretic structural response? Moving average period elongation -10 Elasto-plastic SDOF oscillator with fundamental pre-yielding period: T=0.5s Wavelet Power Spectrum Global Wavelet Spectru Period [s] 0.0315 0.065 0.15 0.5 0.5 1 f y =0.5 T=0.15s T=0.4s T=0.5s 4 8 Yield(red) - SAWP(blue) 0 t=0s Yield Ny=13 Ay=0.38 sec Iy=4.94 nsec - 0 500 1 t=15s Power [(m/s ) ]

10 Acceleration [m/s] fy = 0. T = 0.5 S0 = 1 =0 Sw =0 N /s T input= 0.38 T input1= 0 A [m/s] 5 0 Wavelet analysis of hysteretic response signals -5 How useful CWT is for studying the hysteretic structural response? Moving average period elongation -10 Elasto-plastic SDOF oscillator with Wavelet fundamental Power Spectrum pre-yielding period: T=0.5s Global Wavelet Spectru Period [s] 0.0315 0.065 0.15 0.5 0.5 1 f y =0. T=0.15s T=0.4s T=0.5s 4 8 Yield(red) - SAWP(blue) 0 t=0s Yield Ny=41 Ay=1.6 sec Iy=66.4 nsec - 0 00 400 600 t=15s Power [(m/s ) ]

Wavelet analysis of hysteretic response signals How useful CWT is for hysteretic structural response? Moving resonance does not always occur 6 4 Frequency law - Sw =0.009411 ω= 8π rad/s or T=0.5s k(t) [rad/s] 0 18 16 14 ω= 4π rad/s or T=0.5s 1 t [s] t=0s t=15s

Wavelet analysis of hysteretic response signals How useful CWT is for hysteretic structural response? Moving resonance does not always occur 6 4 Frequency law - Sw =0.009411 0.5 Artificial Accelerogram - Sw=0.009411 ω= 8π rad/s or T=0.5s 0.4 0.3 k(t) [rad/s] 0 18 ddug 0. 0.1 0 16 14 ω= 4π rad/s or T=0.5s -0.1-0. -0.3-0.4 1 t [s] t=0s -0.5 t [s] t=15s

A [m/s] 0 10 0-10 0.0315 0.065 Acceleration [m/s] fy = 0.7 T = 0.5 S0 = 1 =0 Sw =0.009411 N /s T input= 0 T input1= 0 Wavelet analysis of hysteretic response signals How useful CWT is for hysteretic structural response? Moving resonance does not always occur -0 Elasto-plastic SDOF oscillator with fundamental pre-yielding period: T=0.5s Wavelet Power Spectrum Global Wavelet Spec f y =0.7 Period [s] 0.15 0.5 0.5 1 T=0.5s T=0.5s 4 8 Yield(red) - SAWP(blue) 0 t=0s Yield Ny=9 Ay=0.4 sec Iy=.16 nsec - t=15s 0 1 Power [(m/s )

0 Acceleration [m/s] fy = 0.5 T = 0.5 S0 = 1 =0 Sw =0 N /s T input= 0.4 T input1= 0 A [m/s] 10 0-10 Wavelet analysis of hysteretic response signals How useful CWT is for hysteretic structural response? Moving resonance does not always occur -0 Elasto-plastic SDOF oscillator with Wavelet fundamental Power Spectrum pre-yielding period: T=0.5s Global Wavelet Spectru 0.0315 0.065 f y =0.5 Period [s] 0.15 0.5 0.5 1 T=0.5s T=0.5s 4 8 Yield(red) - SAWP(blue) 0 t=0s Yield Ny=17 Ay=0.5 sec Iy=8.84 nsec - t=15s 0 5 10 Power [(m/s ) x ]

10 Acceleration [m/s] fy = 0. T = 0.5 S0 = 1 =0 Sw =0 N /s T input= 0.5 T input1= 0 A [m/s] 5 0-5 Wavelet analysis of hysteretic response signals How useful CWT is for hysteretic structural response? Moving resonance does not always occur -10 Elasto-plastic SDOF oscillator with Wavelet Power fundamental Spectrum pre-yielding period: T=0.5s Global Wavelet Spectru 0.0315 0.065 f y =0. Period [s] 0.15 0.5 0.5 1 T=0.5s T=0.5s 4 8 Yield(red) - SAWP(blue) 0 t=0s Yield Ny=8 Ay=1.59 sec Iy=44.5 nsec - 0 000 4 t=15s Power [(m/s ) ]

Wavelet analysis of hysteretic response signals 8000 Backbone Curve - T=1 =0.05 H=36m Fpr=0.4 APinch=0.6 6000 4000 000 V [kn] 0-000 -4000 T 1 = 0.966s -6000-8000 -.5 - -1.5-1 -0.5 0 0.5 1 1.5.5 Displacement [m] Katsanos/Sextos/Elnashai (014) Ibarra/Medina/Krawinkler (005) model with strength+stiffness degradation

MIP of hysteretic response signals IDA (1 GM is considered) MIPs of input and of output for various IMs

MIP of hysteretic response signals IDA (1 GM is considered) MIPs of input and of output for various IMs

MIP of hysteretic response signals IDA (0 GMs is considered) MIPs after first yielding MIPs near collapse

The angle alpha α of the average MIP MIP is useful but still it is a time-history, while all GM properties and intensity measures (IMs) are scalars We would ideally like to have a wavelet-based scalar quantity to capture the evolving frequency content of GMs Angle alpha α is a scalar!!!

Angle alpha α may not be always positive The angle alpha α of the average MIP

Relation of α with other GM properties: trends and statistics 684 far-field GMs -No pulses; 6.5<M<8.0; 0km<R rup <10km Average value of α increases with PGV (but not so much with PGA) (High PGV values == rich frequency content (presumably towards the end of the GM) == mean frequency content shifts faster from high to low frequencies )

Relation of α with other GM properties: trends and statistics 684 far-field GMs -No pulses; 6.5<M<8.0; 0km<R rup <10km Average value of α increases with T m and decreases with V s,30 (Rich frequency content (presumably towards the end of the GM) == mean frequency content shifts faster from high to low frequencies )

Relation of α with other GM properties: trends and statistics 684 far-field GMs -No pulses; 6.5<M<8.0; 0km<R rup <10km Higher intensity in terms of PGA has a profound impact on the average α trends with T m and V s,30 (for example, higher intensity == soft soils yield == Richer frequency content presumably towards the end of the GM) == mean frequency content shifts faster from high to low frequencies )

Relation of α with other GM properties: trends and statistics 684 far-field GMs -No pulses; 6.5<M<8.0; 0km<R rup <10km

Relation of α with other GM properties: trends and statistics IDA for the previous SDOF system and for the previous 684 far-field GMs using PGA and PGV as IMs Residual analysis for sufficiency of PGA and PGV with respect to α p-value: 0.0096 p-value: 0.0186

Major concluding remarks The CWT is useful and meaningful in studying GMs but care must be exercised in appreciating its limitations (e.g., uncertainty principle) and the fact that there is not a single best wavelet family to use. The MIP seems to be a useful reduction of the CWT in studying the evolution of the mean frequency content of GMs and in capturing the nonlinear behaviour of yielding structures (e.g., moving resonance, period elongation ). The alpha α angle of the MIP appears to be a meaningful scalar to quantify the evolution of the frequency content of GMs and could be used for record selection in PBEE especially to study flexible structures near collapse.

Acknowledgments THANK YOU FOR YOUR ATTENTION Alessandro Margnelli PhD Candidate Dept. of Civil Engineering City University London, London, UK Associate Director at AKTII, London, UK alessandro.margnelli@akt-uk.com Grant Ref: EP/K03047/1 Grant Ref: EP/M01761/1