record of passive surface waves is used that was generated according to Park and

Miller (2005) by considering arbitrary source points around a 2-D (cross) receiver array

with different source intensities (

independent variables, two for the source coordinate (x and y) and one for time (t), to be

denoted as r (x, y, t). Then, FFT is applied to the time axis: R(x, y, w)=FFT[r(x, y, t)],

where w denotes the angular frequency.

Then, for each frequency component within the analyzing range, energy corresponding

to a given azimuth angle for a given phase velocity is calculated according to the

projection principle illustrated in

phase velocities within the analysis range of the two variables (e.g., 5-50 Hz with 0.1-Hz

increments and 10 m/sec-2000 m/sec with 5 m/sec increments ) to construct an

azimuth-energy mapping for one given frequency.

calculated for two arbitrary frequencies of 10 Hz and 23 Hz, respectively. In this map,

azimuths and phase velocities of those passive surface wave events recorded are

recognized from the energy peaks with peak height being proportional to the relative

source intensity. Peaks aligned in the azimuth-axis direction indicate the same mode

(as depicted from the same phase velocity) generated from different locations (as

depicted from different azimuths), whereas those aligned in the phase-velocity-axis

indicate different modes generated from the same location (

Next, all the energy in each mapping is summed (or stacked) together along the

azimuth axis. The azimuth axis is then collapsed and only the phase velocity axis

remains where peaks indicate phase velocities of different modes for the frequency

being analyzed. This stacking has two advantages: 1) the same modes generated

from different locations (different azimuths) are constructively superimposed, and 2) all

the random energy peaks from computational artifacts and unrelated noise wavefields

are relatively suppressed.

All these steps are repeated for different frequencies. Gathering

of these stacked (summed) curves in the order of frequency will

constitute an energy field in two orthogonal axes of phase

velocity and frequency where dispersion trends are recognized

through the energy accumulation patterns (**Fig. 3**).

While the stacking energy in the azimuth-phase velocity space

along the azimuth axis results in dispersion images, the

stacking along the other axis (phase velocity) gives another

useful mapping, called azimuth-frequency space, that shows

energy distribution for different azimuths and frequencies (**Fig. 3**).

Comparing to the conventional wavenumber (f-k) method, the

azimuth scanning approach is more robust and straightforward

(**Fig. 4**).

of these stacked (summed) curves in the order of frequency will

constitute an energy field in two orthogonal axes of phase

velocity and frequency where dispersion trends are recognized

through the energy accumulation patterns (

While the stacking energy in the azimuth-phase velocity space

along the azimuth axis results in dispersion images, the

stacking along the other axis (phase velocity) gives another

useful mapping, called azimuth-frequency space, that shows

energy distribution for different azimuths and frequencies (

Comparing to the conventional wavenumber (f-k) method, the

azimuth scanning approach is more robust and straightforward

(

incorporated in the modeling to generate the 10-sec long field record.

of phase velocities for a surface wave arriving at angle of theta

with a phase velocity of C.

(Below) Fig. 3. Azimuth energy mappings for two arbitrary

frequencies of 10 Hz and 23 Hz, respectively and their stacked

energy along the two orthogonal axes leading to the dispersion

image and azimuth energy spaces.

described above.

e-mail: contact@masw.com |