2020-02-16

Seismic Ambient Noise Notes

目录

1. Notes for Seismic Ambient Noise
   1. Introduce/PSDPDF
   2. Noise Interferometry
   3. Pre- and Post-Process
      1. Introduce
      2. Pre-Process
      3. Post-Process

Notes for Seismic Ambient Noise

Introduce/PSDPDF

1. Natural noise is particularly strong in the microseism band(0.04-0.2Hz)
   p.xv

2. At short periods(0.1-1s)—->human-generated

   intermediate periods(1-30s)—->microseisms(many order higher)

   long-periods(30-500s)ocean infragravity waves

3. Short-period. Attenuate within several kilometers in distance and depth.

   Cultural Noise. often bimodal due to diurnal variations in human activities.

   Glacier Calving.

   Wind

4. Mid-period

   Microseisms. unaffected by common anthropogenic noise sources. Strong surface wind—>ocean swell—>microseisms energy. Whether is generated in shallow water or deep is still a question. Can propagate hundreds to thousands of kilometers.

   Extra- Tropical Storms/Tropical Cyclones.(非热带风暴/热带气旋). short and mid periods are increase but low in long-periods. earthquake signals generate high long-period surface wave power.

   Sea Ice. In winter, sea ice dampens ocean wave heights and short-period microseism power is reduced

5. Long-period. In general, power in long-period bands greater than 20 s is of interest to the seismology community for earthquake surface waves and long-period resonances in the solid Earth. A less common but useful signal in this band are surface waves generated by large calving glaciers in Greenland. Often, optimal noise levels in the long period are difficult to record since it requires both a high-quality broadband sensor and a well-built seismic station.

   Hum. due to portion of ocean swell reaches coastlines and transform into much longer periods(>50s). climate-related and seasonal and decadal cycles.

   Seiche.

6. Application of PSDPDF.

   Seismic station design and construction.

   Quality assessment of seismic data or metadata

Noise Interferometry

• equipartition: all propagation modes are equally strong and statistically uncorrelated
  arise either directly through the action of uncorrelated and homogeneously distributed noise sources or indirectly through sufficiently strong multiple scattering.

• wavefields are not generally equipartitioned for scattering may be too weak and attenuation too strong distribution of noise sources is strongly heterogeneous and time-variable

• Frequency-dependent arrival times of fundamental-mode surface waves are empirically found to be rather robust

• Normal-mode Summation: Request all modes have different frequency. Cross-terms from modes with same frequency will cancel out when sources are uncorrelated over long timescales.

• For example, modes on Earth are certainly never close to equipartitioned for at least two reasons. For one, the majority of noise sources on Earth are expected to be at or near the Earth’s surface (ocean waves, wind, industrial activity, traffic) and such surface sources excite fundamental-mode waves much more strongly than they excite higher-order modes. Moreover, even within the subset of fundamental-mode waves, the fact that large noise sources, like ocean storms, are concentrated spatially in certain areas of the Earth implies that waves from certain directions will be much stronger and will therefore not be close to equipartitioned.

• Plane Wave: Using surface wave Green function and derivate the same result. When A(θ) is a constant, then the integral can be calculate accurately. When not,can use stationary phase point at 0(positive) and 180(negative). And 2D plane wave methods can assess the degree to which a non-uniform noise distribution by consider the stationary-phase approximation or use numerically calculation. Travel-times are only affect the second order.

• Representation Theorems:

• Discussion:

  Green’s Function Retrieval is successfully used dues to its simplicity. Though the theoretical requirements of Green’s function retrieval are generally not met on Earth, at least the frequency-dependent traveltimes (dispersion) of fundamental-mode surface waves are empirically reliable and for most applications sufficiently relatable to the traveltimes of the Green’s function.

  Drawback: Because of the complexities of the Earth, the traveltimes, amplitudes, and waveforms may be incorrect. Thus, exploit complete waveforms for improved resolution are therefore not applicable.

Pre- and Post-Process

Introduce

• a heuristic explanation by hyperbolas. End-fire direction ->constructive. Enormous event or series of smaller event occur persistently at a single location(26s microseism, Kyushu microseism)->pernicious
• As long as there are events in the end-fire direction, a reliable estimated Green Function will emerge eventually. It’s just a matter if observational time.
• SNR is the divide between the peak amplitude and RMS of the trailing noise.
• Precursory noise is generated at the zero lag time. Signal of interest will be the latest signal. And the trailing noise is followed by the useful signal
• RMS is root mean square, which is represented by

$$RMS=\sqrt{\frac 1n \sum_{i=1}^n x^2_i}$$

• complex conjugate in frequency domain is correlation in time domain
• Ambient noise propagates over long distance typically originates in the oceans

• Two data ->remove instruments response ->remove mean ->remove trend ->band pass

• For slower rate to cross-correlation, time domain weight functions are used. The downside is the loss of the meaningful amplitude information.

• For those who are interested in amplitude, binary time domain weighting scheme can be used: 0 for time periods of earthquakes or IR and 1 for other periods

Pre-Process

Post-Process

• sign function:

  $$sgn(x)= \begin{cases} 1, &if\ x>0\\ 0, &if\ x=0\\ -1,&if\ x<0 \end{cases}$$

• Hilbert transform is just a convolution with

$$h(t)=\frac 1\pi \int_{-\infty}^\infty \frac {x(t)}{t-\tau}d\tau$$

• Surface Wave Dispersion

  1. correlation function s(t) Fourier transform:

     $$S(\omega) = \int_{-\infty}^\infty s(t)exp(i\omega t)dt$$

  2. Analytic signal:

     $$S_a (\omega) = S(\omega)(1+sgn(\omega)) \\ s_a(t) = s(t)+ih(t) = A(t)exp(i\phi(t))$$

  3. Gaussian filter with center frequencies $ \omega _0 $:

     $$S_a(\omega,\omega_0) = S(\omega)(1+sgn(\omega))G(\omega-\omega_0) \\ G(\omega-\omega_0) = exp(-\alpha(\frac {\omega-\omega_0}{\omega_0})^2)$$

     where $ \alpha$ is tunable parameter that defines the complementary resolutions in the time and frequency domain and is commonly made range dependent.

  4. Inverse Fourier transforming each band-passed function $ S_a(\omega,\omega_0) $ back to the time domain yields the 2D envelope function, $ A(t,\omega_0)$ and phase function $ \phi(t,\omega_0)$. Group velocity is measured from $ A(t,\omega_0)$, phase velocity from $ \phi(t,\omega_0)$.

  5. Group velocity:

     $$U(\omega) = \frac r{t_g},where\ r \ is\ interstation\ distance$$

     $ t_g$ is measured using the peak of the envelope function at each center frequency. If the speed changes rapidly with center frequency. Center frequency be replaced by the so-called “instantaneous frequency” $ \omega = [\partial \phi(t,\omega_0)/\partial t]_{t=t_g}$.

  6. Phase velocity:

     $$\phi(t,\omega)=kr-\omega t+\pi/2-\pi/4+n\cdot2\pi \\ c=\frac \omega k=\frac {r\omega}{\phi(\omega,t_g)+\omega t_g-\pi/4-n\cdot2\pi}$$

  7. n is more easily resolved at longer periods, and once it is known for a station-pair at any frequency it is known for all frequencies. Phase speed is typically determined more reliably than group speed.

  8. Because of the far field approximation, in time domain, interstation that distance less than 2 wavelengths will be discarded or frequency-domain dispersion measurements.

  9. Methods like time-domain normalization and frequency whitening methods presented here are not linear, which means the order of application of the time-domain and frequency-domain filters matters.

• Continental Love Waves

  1. In early research， ambient noise was applied predominantly to vertical component seismic records to recover fundamental mode Rayleigh waves. It’s now understood that Love waves are well represented in ambient noise and they provide important information about both crustal and uppermost mantle anisotropy through a number of studies.

  2. Principal differences in processing ambient noise data for Love waves compared to Rayleigh waves:

     • First, to retain meaningful geometrical information upon rotation transverse to the interstation direction, the East and North component records for each station must be normalized identically (in both time and frequency) prior to cross-correlation. (Lin et al.)
     • Second, cross-correlations are computed between all components. Given the three components of the seismograms at two stations (Z - vertical,E - East, N - North), nine cross-correlations can be performed, ZZ, EE, NN, EN,NE, ZN, NZ, ZE, and EZ.
     • Finally, the cross-correlations are rotated into the Radial (R) and Transverse (T) components,
       each of which point in the same direction at both stations. Rotating after cross-correlation is computationally more efficient than rotating before cross-correlation.
  3. Rayleigh waves are observed at both positive and negative lags on the ZZ and RR components. Love waves, which are faster than Rayleigh waves in this period band, appear on the TT component. Rayleigh waves are stronger on the positive lag and Love waves on the negative lag, which means that the azimuthal content of Rayleigh and Love waves differ. The relative lack of longer-period Love wave dispersion measurements probably results mostly from the higher noise levels on horizontal components.

学术

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