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A review of time‐frequency methods
The 2014 sponsors meeting focused on time-frequency analysis (TFA) methods for microseismic applications, presenting new algorithms and their advantages over traditional techniques. Key applications include resonance frequency analysis, event separation, and improved signal-to-noise ratios. The meeting emphasized developing a comprehensive set of tools for microseismic spectral analysis, highlighting challenges in wave separation and the potential for new research collaboration.
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A review of time‐frequency methods
- 1. Sponsors Meeting 2014 Areview of time‐frequency methods with application to body-wave separation Roberto Henry Herrera, Jean-Baptiste Tary and Mirko van der Baan* University of Alberta, Canada Microseismic Industry Consortium
- 2. Objective – Valueproposition Sponsors Meeting 2014 • Objective: – Review of best performing techniques for time-frequency analysis • Present our home-brewed algorithms with their recipes. • Possible applications: – Resonance frequency analysis & LP events. – Represent sharp events. Short duration and low energy. – Separate out close events in time and close frequency components. • Main problem • Latest review of TFA is from the past century (20 years ago). • Many new methods but hard to find best suited for specific problems. • Push the limits of the Gabor uncertainty principle. • Value Proposition • “a comprehensive set of essential tools for microseismic spectral analysis”. • Separation via differences in freq content. Requires hi-res time-freq transforms. Reconstruct P and S waves from the time-frequency map.
- 3. Sponsors Meeting 2014 TFA a cornerstone in geophysical signal processing and interpretation. Why are we going to the T-F domain? Study changes of frequency content of a signal with time. Useful for: - attenuation measurement (Reine et al., 2009) - direct hydrocarbon detection (Castagna et al., 2003) - stratigraphic mapping (ex. detecting channel structures) (Partyka et al., 1998). - Microseismic event detection (Das and Zoback, 2011) Extract sub-features in seismic signals - reconstruct band‐limited seismic signals from an improved spectrum. - improve signal-to-noise ratio of the attributes. (Steeghs and Drijkoningen, 2001). - identify resonance frequencies (microseismicity). (Tary & van der Baan, 2012). Time-Frequency Analysis (TFA)
- 4. Sponsors Meeting 2014 Motivation:The last 10-15 years have seen the development of many new high-resolution decompositions Fourier and Wavelet Transforms. The “workhorses” of spectral analysis Methods 1. Short-time Fourier Transform (STFT) 2. Continuous Wavelet Transform (CWT) 3. Stockwell Transform (ST) 4. Matching Pursuit (MP) 5. Synchrosqueezing Transform (SST) 6. Short-time Autoregressive (ST-AR) 7. Kalman Smoother (KS) 8. Empirical mode decomposition (EMD) Benchmark signals 1. A Toy Example – Synthetic signal. 2. A laughing voice. 3. A volcano tectonic event – Gliding tremor. (Redoubt Volcano on March 31, 2009). 4. A microseismic event. (Rolla HyFrac. 2011) 5. And a global earthquake signal (Tohoku 2011, Mw9) “A comprehensive set of essential tools for microseismic spectral analysis” The review: Chapter 2: Spectral estimation – What’s new? What’s next?
- 5. Sponsors Meeting 2014 Arepresentative volcano-seismic signal Gliding tremor: Redoubt Volcano on March 31, 2009. Some volcanoes 'scream' at ever-higher pitches until they blow their tops. http://www.sciencedaily.com/releases/2013/07/1307 14160521.htm Hotovec et al., 2013, Strongly gliding harmonic tremor during the 2009 eruption of Redoubt Volcano.Journal of Volcanology and Geothermal Research, 2013; 259: 89. Redoubt Volcano’s active lava. Dome. Alaska. Credit: Chris Waythomas, Alaska Volcano Observatory Swarms of small earthquakes can precede a volcanic eruption, sometimes resulting in "harmonic tremor" resembling sound from some musical instruments.
- 6. Sponsors Meeting 2014 Aglobal seismology example: Megathrust earthquake: Tohoku-Oki, March 11, 2011, Mw9 STFT SSTCWT MPST ST-AR KSCEEMD The seismogram was recorded by the borehole station KDAK from the IRIS IDA network located in Kodiak Island on the Aleutian trench, South Alaska.
- 7. Sponsors Meeting 2014 Data:hydraulic fracture treatment, western Canada. Rolla, BC, 2011. Field layout. Eaton et al. (GJI, 2014)
- 8. Sponsors Meeting 2014 Microseismicevent – Rolla, BC, 2011. STFT P-S converted wave - 320 Hz S-wave - 210 and 320 Hz. Signal TFR - is challenging very short duration events (0.1 - 1 s). A clear separation of seismic phases is difficult to obtain due to the limits in time and frequency resolutions of conventional T-F methods. Microseismic event Mw -1.7. Vertical component, deepest geophone.
- 9. Sponsors Meeting 2014 Microseismicevent – TFT with 8 methods +++ Smearing + Smearing + Smearing STFT SSTCWT MPST ST-AR KSCEEMD --- Smearing -- Smearing - Smearing - Smearing +- Smearing
- 10. SST – Steps Synchrosqueezingdepends on the continuous wavelet transform and reassignment Sponsors Meeting 2014 Microseismic signal 𝑠(𝑡) Mother wavelet 𝜓(𝑡) 𝑓, Δ𝑓 CWT 𝑊𝑠(𝑎, 𝑏) IF 𝑤𝑠 𝑎, 𝑏 Reassignment step: Compute Synchrosqueezed function 𝑇𝑠 𝑓, 𝑏 Extract dominant curves from 𝑇𝑠 𝑓, 𝑏 + ICWT Time-Frequency Representation Signal Reconstruction - Sum of modes - Selected areas
- 11. Continuous Wavelet Transformvs Synchrosqueezing Transform Sponsors Meeting 2014 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -1 0 1 2Amplitude Synthetic trace s(t) CWT SST 100 Hz 30 Hz 7 Hz 30 Hz 40 Hz 20 Hz 20 Hz 20 Hz Morlet atom 100 Hz
- 12. Single-station separation ofP- & S-waves? Sponsors Meeting 2014 • Objective: – Can we separate P & S waves at a single station w/o prior knowledge about polarities or waveforms? • Option 1: Separation of P & S waves via curl and divergence => Requires closely spaced multiple stations • Option 2: Separation via differences in freq content => Requires hi-res time-freq transforms
- 13. Microseismic event STFT & SST 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.5 0 0.5 1 x 10 4 Amplitude Time (s) s(t) S-wave P-wave P-wave S-wave Sponsors Meeting 2014
- 14. Sponsors Meeting 2014 Polarization+ Move-out Analysis East North Vertical P-wave Sh-wave Sv-wave P- to Sv-wave a)- 3C microseismic traces at geophone 7, stage 2. b)- Polarization vectors for the waves modes. The polarity is reversed for display purposes only. East North Vertical
- 15. Sponsors Meeting 2014 Phaseidentification: Move-out Vertical Component. Ringing P-wave arrival Picking of P-to-S wave P-wave Sh-wave Sv-waveP to Sv-wave 2 3 5 7 Very similar move- outs. - 2 wave packets for P-waves picks, w/ similar apparent velocities but different polarizations. (P + P-to-S waves) - 2 S-waves w/ slightly different: apparent velocities, arrival times and polarizations. - The “fast” S-wave on the East-North components is the Sh and the “slow” S-wave on the vertical is the Sv. Move-out analysis compatible with the results of the analysis of the time series & polarizations.
- 16. Sponsors Meeting 2014 Microseismicevent – Rolla, BC, 2011. a)- Hodograms for the stage 2 event. b)- Vectors corresponding to the hodograms. P-wave Sh-wave Sv-wave P- to Sv-wave
- 17. Sponsors Meeting 2014 Projection& Time Frequency Representation 320 210 ~210 ~300 215 320 320 200 260 ~230 P Sv Sh AmplitudeAmplitudeAmplitude
- 18.
- 19. Sponsors Meeting 2014 P-waveS-wave 0.325 0.33 0.335 0.34 0.345 0.35 200 250 300 350 Frequency(Hz) 0.325 0.33 0.335 0.34 0.345 0.35 -2000 0 2000 Amplitude Time(s) s(t) sr (t) 0.36 0.365 0.37 0.375 0.38 0.385 0.39 200 250 300 350 Frequency(Hz) 0.36 0.365 0.37 0.375 0.38 0.385 0.39 -1 0 1 x 10 4 Amplitude Time(s) s(t) sr (t) Signal extraction from time-freq map
- 20. Conclusions Sponsors Meeting 2014 SST: •High-resolution time-frequency decomposition – Attractive for detailed analysis of variety of signals • Microseismic + earthquake data, any other signals • SST also permits signal reconstruction: – SST can extract individual components (= time-varying monochromatic signals) – Sum of individual components ≈ original signal • Very acceptable reconstruction error • We are developing a complete toolset for High- Res TFA.
- 21. Acknowledgments Sponsors Meeting 2014 •Sponsors of the Microseismic Industry Consortium for financial support • David Eaton: – For providing microseismic data • Sergey Fomel: – For many encouraging discussions on SST and P/S wave separation • Melanie Grob and Shawn Maxwell: – For their interesting suggestions that helped to improve the interpretation of our results.
- 22. Conclusions Sponsors Meeting 2014 SST: •High-resolution time-frequency decomposition – Attractive for detailed analysis of variety of signals • Microseismic + earthquake data, any other signals • SST also permits signal reconstruction: – SST can extract individual components (= time-varying monochromatic signals) – Sum of individual components ≈ original signal • Very acceptable reconstruction error • We are developing a complete toolset for High- Res TFA.
- 23. Rolla Experiment. StageA2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.5 0 0.5 1 x 10 4 Amplitude Time (s) s(t) S-wave P-wave P-wave S-wave
- 24. Rolla Experiment. ModeDecomposition 0.2 0.4 0.6 0.8 1 -1 0 1 x 10 4 Original trace Amplitude 0.2 0.4 0.6 0.8 1 -5000 0 5000 Mode 1 Amplitude Time (s) 0.2 0.4 0.6 0.8 1 -6000 -4000 -2000 0 2000 4000 Mode 2 Amplitude Time (s) 0.2 0.4 0.6 0.8 1 -1000 0 1000 Mode 3 Amplitude Time (s) 0.2 0.4 0.6 0.8 1 -500 0 500 Mode 4 Amplitude Time (s) 0.2 0.4 0.6 0.8 1 -1 0 1 Mode 5Amplitude Time (s)
- 25. Signal extraction fromRolla Stage A2 P-wave S-wave 0.325 0.33 0.335 0.34 0.345 0.35 200 250 300 350 Frequency(Hz) 0.325 0.33 0.335 0.34 0.345 0.35 -2000 0 2000 Amplitude Time(s) s(t) sr (t) 0.36 0.365 0.37 0.375 0.38 0.385 0.39 200 250 300 350 Frequency(Hz) 0.36 0.365 0.37 0.375 0.38 0.385 0.39 -1 0 1 x 10 4Amplitude Time(s) s(t) sr (t)
- 26. P-wave SH? 0.325 0.33 0.3350.34 0.345 0.35 290 300 310 320 330 Frequency(Hz) 0.325 0.33 0.335 0.34 0.345 0.35 -2000 0 2000 Amplitude Time(s) s(t) sr (t) 0.36 0.37 0.38 0.39 0.4 0.41 180 200 220 240 Frequency(Hz) 0.36 0.37 0.38 0.39 0.4 0.41 -1 0 1 x 10 4 Amplitude Time(s) s(t) sr (t) 0.37 0.38 0.39 0.4 280 300 320 340 Frequency(Hz) 0.37 0.38 0.39 0.4 -1 0 1 x 10 4 Amplitude Time(s) s(t) sr (t) SV? Signal extraction from Rolla Stage A2
- 27. 0.3 0.32 0.340.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 150 200 250 300 350 Frequency(Hz) 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 -1 0 1 x 10 4 Amplitude Time(s) s(t) sr (t) Signal extraction from Rolla Stage A2
- 28. Rolla Experiment. WellA Stage 3 Fig 11. Eaton et. al., 2014 0 0.5 1 1.5 -2 0 2 x 10 -7 Amplitude Time(s) s(t) S-wave P-wave P S Time-Frequency Rep. by SST Scattered waves?
- 29. Rolla Experiment. WellA Stage 3 0.7 0.705 0.71 0.715 0.72 0.725 0.73 200 250 300 350 Frequency(Hz) 0.7 0.705 0.71 0.715 0.72 0.725 0.73 -0.2 -0.1 0 0.1 Amplitude Time(s) s(t) s r (t) P-wave 0.75 0.76 0.77 0.78 0.79 150 200 250 300 Frequency(Hz) 0.75 0.76 0.77 0.78 0.79 -0.5 0 0.5 Amplitude Time(s) s(t) sr (t) S-wave
- 30. Rolla Experiment. StageA3 East North Vert. P-wave Sh-wave Sv-wave Vectors corresponding to the hodograms The three phases P, Sv, and Sh are approximately mutually perpendicular.
- 31. Rolla Experiment. StageA3 P-wave Sh-wave Sv-wave 3C data projected on P vector SST 510 Hz400 Hz 270 Hz 290 Hz 195 Hz - P-waves at 400 Hz - Remnants of P-Sv converted waves at 270 Hz? - Difficulties to separate P- and Sv-waves - Sv contributions at 290 Hz (see next slide) - Patch at ~195-200 Hz present in all components - Patch at 510 Hz ?
- 32. Rolla Experiment. StageA3 P-wave Sh-wave Sv-wave 3C data projected on Sv vector SST 455 Hz330 Hz 225 Hz 310 Hz 210 Hz - Sv-waves at 310 Hz - P-Sv converted waves at 330 and 225 Hz? - Patch at 455 Hz?
- 33. Rolla Experiment. StageA3 P-wave Sh-wave Sv-wave 3C data projected on Sh vector SST 350Hz 295 Hz 190 Hz - Sh-waves between 295 and 350 Hz
- 34. Rolla Experiment. StageA2 3C data projected on vectors SST P Sv Sh
- 35. Questions TLE. 2012. BradBirkelo et. al. 1- Are the two components of the P-wave related to a compensated linear vector dipole (CLVD), instead of a double couple (DC) fracture type?. 1a)- CLVD a possible mechanism for microseismic fractures (Baig, A., and T. Urbancic ,2010) 2- We are able to extract regions on the Time-Freq map. Do you envision any application of waveform separation in microseismic analysis? 3- Is full-waveform based moment tensor inversion a possible application? 4- We would appreciate your collaboration in future related work, what are the main challenges you would like to work on?
- 36. Review Paper Signalsand TFR
- 37. Review Paper Signalsand TFR
Editor's Notes
- #3 Title - Latest developments in time-frequency analysis Time-frequency analysis is a cornerstone in geophysical signal processing and interpretation. The last 10-15 years have seen the development of many new high-resolution decompositions that are often fundamentally different from Fourier and wavelet transforms. This proposed 1/2-day PCWS intends to invite algorithm developers to explain their methods and show results on a provided collection of benchmark signals, including a laughing voice, a volcano tremor, a microseismic event and a global earthquake, with the intention to provide a fair comparison of the pros and cons of each method.
- #5 This proposed 1/2-day PCWS intends to invite algorithm developers to explain their methods and show results on a provided collection of benchmark signals, including a laughing voice, a volcano tremor, a microseismic event and a global earthquake, with the intention to provide a fair comparison of the pros and cons of each method. 3- Gliding tremor: Redoubt Volcano on March 31, 2009. Frictional-faulting model for harmonic tremor before Redoubt Volcano eruptions. Nature Geosciences. 2013. Hotovec-Ellis is a co-author of a second paper, published online July 14 in Nature Geoscience, that introduces a new "frictional faulting" model as a tool to evaluate the tremor mechanism observed at Redoubt in 2009. The lead author of that paper is Ksenia Dmitrieva of Stanford University, and other co-authors are Prejean and Eric Dunham of Stanford. Read more at: http://phys.org/news/2013-07-volcanoes-ever-higher-pitches-tops.html#jCp Some volcanoes 'scream' at ever-higher pitches until they blow their tops 4- The fracturing is monitored by a toolstring of 6 downhole short-period geophones, with a sampling frequency of 2000 Hz. The signal shown in Figure 8 corresponds to the recordings of a magnitude -1.7 microseismic event by the vertical component of the deepest geophone. 5- The Mw9 Tohoku earthquake occurred on March 11, 2011
- #6 This proposed 1/2-day PCWS intends to invite algorithm developers to explain their methods and show results on a provided collection of benchmark signals, including a laughing voice, a volcano tremor, a microseismic event and a global earthquake, with the intention to provide a fair comparison of the pros and cons of each method. 3- Gliding tremor: Redoubt Volcano on March 31, 2009. Frictional-faulting model for harmonic tremor before Redoubt Volcano eruptions. Nature Geosciences. 2013. Hotovec-Ellis is a co-author of a second paper, published online July 14 in Nature Geoscience, that introduces a new "frictional faulting" model as a tool to evaluate the tremor mechanism observed at Redoubt in 2009. The lead author of that paper is Ksenia Dmitrieva of Stanford University, and other co-authors are Prejean and Eric Dunham of Stanford. Read more at: http://phys.org/news/2013-07-volcanoes-ever-higher-pitches-tops.html#jCp Some volcanoes 'scream' at ever-higher pitches until they blow their tops The Mw9 Tohoku earthquake occurred on March 11, 2011, offshore the North-West coast of Japan where the Pacific plate subducts under the Okhotsk plate (Tajima and Kennett, 2012). The seismogram presented in Figure 10 was recorded by the borehole station KDAK from the IRIS IDA network located in Kodiak Island on the Aleutian trench, South Alaska. This station has a 3-component broadband seismometer with sampling frequency at 20 Hz giving an usable frequency band between around 0.003 and 10 Hz. The main signal from the earthquake lasts for about half an hour, with a high SNR superior to 10.
- #7 The Mw9 Tohoku earthquake occurred on March 11, 2011, offshore the North-West coast of Japan where the Pacific plate subducts under the Okhotsk plate (Tajima and Kennett, 2012). The seismogram presented in Figure 10 was recorded by the borehole station KDAK from the IRIS IDA network located in Kodiak Island on the Aleutian trench, South Alaska. This station has a 3-component broadband seismometer with sampling frequency at 20 Hz giving an usable frequency band between around 0.003 and 10 Hz. The main signal from the earthquake lasts for about half an hour, with a high SNR superior to 10. The magnitude 9.0 Tohoku-Oki, Japan, earthquake on 11 March 2011 is the largest earthquake to date in Japan’s modern history and is ranked as the fourth largest earthquake in the world since 1900.
- #9 The signal shown in this Figure corresponds to the recordings of a magnitude -1.7 microseismic event by the vertical component of the deepest geophone. The T-F representation of these events is complicated mainly by their very short durations generally between 0.1 and 1 s. A clear separation of the different seismic phases is then difficult to obtain due to the limits in time and frequency resolutions of conventional T-F methods.
- #10 Explain each method. Time-frequency representations of the microseismic event in Figure 8), using the ST, ST-AR, KS, CWT, SST, EMD and MP. The computing parameters are, ST: resolution parameter k=4; ST-AR: window of 0.04 s with 90 % overlap and an AR order of 11; KS: AR order of 11; CWT and SST: Morlet wavelet with 64 voices per octave; CEEMD: 100 realizations and random noise injection with 15 % of the signal maximum amplitude; MP: Gabor dictionary.
- #12 The challenging synthetic signal: - 20 Hz cosine wave, superposed 100 Hz Morlet atom at 0.3 s two 30 Hz zero phase Ricker wavelets at 1.07 s and 1.1 s, three different frequency components between 1.3 s and 1.7 s of respectively 7, 30 and 40 Hz.
- #14 Time-frequency representations of the microseismic event in Figure 8), using the ST, ST-AR, KS, CWT, SST, EMD and MP. The computing parameters are, ST: resolution parameter k=4; ST-AR: window of 0.04 s with 90 % overlap and an AR order of 11; KS: AR order of 11; CWT and SST: Morlet wavelet with 64 voices per octave; CEEMD: 100 realizations and random noise injection with 15 % of the signal maximum amplitude; MP: Gabor dictionary.
- #15 Small ringing problem for this geophone. Especially Vert.
- #16 Small ringing problem for this geophone. Especially Vert. Taking all the first P-wave picks, the Vp apparent is ~11810 m/s The East comp. gives the right Vp/Vs ratio, being different from the theoretical 1.73 by 15%. This includes errors in time-picks, effects from non-ideal rocks and slightly different ray paths between P- and S-waves. It seems there are 2 wave packets for P-waves picks, having similar apparent velocities but different polarizations. (P + P-to-S waves) Similarly, two S-waves are present. They have slightly different apparent velocities, different arrival times and different polarizations. The “fast” S-wave on the East-North components is the Sh and the “slow” S-wave on the vertical is the Sv. The measurements of apparent velocities can change quickly even for small errors in time-picks. Delay between P and P-to-Sv wave ranging between 0.002 and 0.0085 s Considering a S-wave velocity of 5500/1.73 ~ 3180 m/s, and with delay Dt = distance*(1/Vp – 1/Vs), it gives distances between 15 and 64 m So conversion likely in the hosting rock close to the borehole.
- #17 Small ringing problem for this geophone. Especially Vert. Taking all the first P-wave picks, the Vp apparent is ~11810 m/s The East comp. gives the right Vp/Vs ratio, being different from the theoretical 1.73 by 15%. This includes errors in time-picks, effects from non-ideal rocks and slightly different ray paths between P- and S-waves. It seems there are 2 wave packets for P-waves picks, having similar apparent velocities but different polarizations. (P + P-to-S waves) Similarly, two S-waves are present. They have slightly different apparent velocities, different arrival times and different polarizations. The “fast” S-wave on the East-North components is the Sh and the “slow” S-wave on the vertical is the Sv. The measurements of apparent velocities can change quickly even for small errors in time-picks. Delay between P and P-to-Sv wave ranging between 0.002 and 0.0085 s Considering a S-wave velocity of 5500/1.73 ~ 3180 m/s, and with delay Dt = distance*(1/Vp – 1/Vs), it gives distances between 15 and 64 m So conversion likely in the hosting rock close to the borehole.
- #19 Frequency content analysis Similar frequency components on the two geophones. The P-wave and the P-to-Sv waves have similar frequency contents (320 Hz) and their arrival times are very close. Despite its high-resolution the SST is unable to clearly separate them. They are somewhat distinguishable by their polarization. The Sv-wave consists of two components located at 210 and 280-340 Hz. The tails that are observed on geophone 7 only could be related to instrument self-excitation. The Sh-wave has two components, one at 200-215 and one at 260-275 Hz. The component at 200-210 Hz for all waves might correspond to energy leakage between components due to non-linear polarization or to instruments ringing.
- #25 Zoom in of the area in the rectangles shows a better P and S wave separation. A global mode extraction from the Wavelet’s ridges drives to the next slide.
- #26 Mode decomposition: useful of noise suppression and extraction of individual components. This is unsupervised mode extraction. Only Mode 2 seems to separate P and S wave, which is useful if we want on a clean time domain signal. But if we manually extract the Region of Interest (ROI), then the next slide shows promising results.
- #27 Extracting a vertical rectangle comprising only P wave Time-Frequency component drives to a P wave only reconstruction. The same with the rectangle gathering S wave spectral content. The bottom plots shows in Blue the actual P-wave (left) and S-wave (right) and in red their corresponding reconstructed P and S waves. Something interesting is that both P and S waves are built of two frequency components. Extracting just one of these components give a different time domain signal. And this observation brings more questions than answers to our research.
- #28 This slide can be deleted to save time.
- #29 This is a hidden slide just in case some one wants to see to reconstruction from the entire section.
- #30 Change converted waves by scattered waves. Eaton, D., Baan, M. van der, Tary, J., & Birkelo, B. (2013). Broadband microseismic observations from a Montney hydraulic fracture treatment, northeastern BC, Canada. Recorder, (3). Retrieved from http://csegrecorder.com/articles/view/broadband-microseismic-observations-from-a-montney-hydraulic-fracture-treat Examples of high-frequency microseismic events, which are generally dominated by S arrivals. Upper panels: signal and noise spectra. Lower panels: recorded waveforms and corresponding spectrograms obtained using the short-time Fourier transform. Waveform examples of high-frequency microseismic events are presented in Figure 4. In this case, the S-wave arrival has the highest amplitudes. The frequency content of these signals is predominantly concentrated above 100 Hz, and the bandwidth decreases with distance due to the effects of attenuation. Again we see Two frequency components for the P waves and two Freq components for the S wave.
- #31 Again we see two frequency components for the P waves and two freq components for the S wave. If we take the components the reconstruction is almost exact to the original waveform.
- #37 Can maybe be deleted.