Next: Characterization of PP phases
Seismic arrays are able to measure the properties of propagating waves. The
arrays were built to detect the origin of seismic signals. This can be done
by determining the vector velocity of a seismic wave. The vector velocity can
be separated into the two components, horizontal slowness (i.e. the
inverse of the apparent horizontal velocity) and backazimuth .
Several methods to measure slowness and backazimuth with array data have
been developed (Davies et al. 1971; Capon, 1973; Posmentier and
Herrmann, 1971). In this study, the frequency-wavenumber analysis
(fk-analysis) is used. The fk-analysis is a standard array technique
which simultaneously calculates the power distributed among different
slownesses and directions of approach (Capon, 1973; Harjes and Henger,
1973; Aki and Richards, 1980).
Most array methods use time delays of the signals recorded at different
array stations and summation of the coherent data stream from each station to
improve the signal-to-noise ratio (SNR) for a specific phase. The time
delays required to bring the signals into phase provide a direct estimate
of the backazimuth and the slowness of the signal. If the slowness and
backazimuth of a signal are unknown, a grid search for all u and
combinations can be performed to find the best parameter combination,
producing the highest amplitudes of the summed signal. This computation
is performed in the spectral domain due to lower computational time.
The following derivation of the fk-analysis follows Kelly (1967) and
Harjes and Henger (1973).
A signal arriving at a reference point within the array with a horizontal
velocity, , and a backazimuth, , can be written as . The
n-th seismometer with the location vector,
, relative to
the array reference point records the signal, :
is the slowness vector with
The maximum amplitude of the sum of all array seismometers is reached if
the signals of all stations are in phase, i.e. if the time shifts
The output of the array can be computed by
for an array of N elements. For a signal with a different slowness vector
the beam trace is computed using equation (5.1):
The total energy recorded at the array can be calculated by the integration
of the squared summed amplitudes over time
using Parseval's theorem. In equation (5.5), S) is the Fourier
transformation of and k is the wavenumber vector with
the wavenumber vector for
The backazimuth determines the direction of and the slowness
the magnitude of .
Equation (5.5) can be written as:
as array response function (ARF).
As can be seen from equation (5.7), the total energy recorded at the array is
defined by the power spectral density
and the ARF
. The ARF is controlled by
the design (aperture, configuration and interstation spacing) of the array.
The result of the fk-analysis is a power spectral density as a function of
slowness and backazimuth. The slowness can be calculated from the wavenumber
with as the apparent horizontal slowness.
The backazimuth can be calculated by
The power spectral density is displayed in a polar-coordinate system
called the fk-diagram. In the fk-diagram, the backazimuth is plotted
on the azimuthal axis and the slowness on the radial axis.
Figure 5.2a) shows the array response of Yellowknife. The ARF was computed for
a monochromatic wave with a frequency of 1 Hz arriving at an array with
YKA configuration. It has been assumed that the power spectral density
which leads to
where the wave arrives with a slowness u = 0, i.e.
= 0. The
logarithmic power is colour coded and the maximum is normalized to 0 dB.
The lines are -1 dB contour isolines. The slowness and backazimuth of
the maximum power is marked by the white circle. The maximum slowness
displayed is 12 s/ with 2 s/ tick intervals. The azimuth
describes the full circle from 0 to 360 clockwise.
The cross-shaped form of the ARF resembles the geometry of YKA and is
important for the detection of PP as discussed in chapter 5.2.1.
Figure 5.2b) shows the logarithmic colourscale valid for all fk-diagrams
displayed in this thesis.
response function of YKA computed for 1 Hz data. The logarithmic power
spectral density is displayed as colour and -1 dB isolines are added. The
maximum power density is normalized to 0 dB (red). The slowness with 2
s/ per tick is displayed on the radial axis, the backazimuth
clockwise from 0 to 360. The maximum power is marked by
the white circle.
b) Colour scale for all fk-diagrams in this thesis.
c) Same as a) but for a synthetic monochromatic (1 Hz) sinusoidal wave
arriving at YKA with a slowness 7.5 s/ and backazimuth
d) Same as a) but for a P arrival of real data (event: 04-jun-1993 10:49) at
YKA. Theoretical slowness and backazimuth values are marked by the white
diamonds. The lower slowness marks the P slowness (u = 4.43). The other
diamond marks the PP slowness (u = 7.65). The theoretical bazimuth is
For coherent waves with 1 s period arriving from different backazimuths
and with slownesses different from u = 0 s/, the maximum of
the power spectral density is shifted to the corresponding values, without
changing the form of the ARF. This is shown in Figure 5.2 c) for a signal
arriving with a slowness of u = 7.5 s/ and a backazimuth of
225. For simplicity, less isolines are displayed in this figure than
in Figure 5.2 a). For a signal with different frequency f
f = 1 Hz the fk-diagram is scaled by the factor f/f
without a change of the cross shaped form of the ARF.
Real seismograms do not consist of monochromatic waves, but of wavelets showing
a broad frequency spectrum. For these signals the fk-diagram is computed
as an integration over all frequencies in a chosen frequency band.
The cut-off frequencies for the fk-analysis coincide with the edges of the
used band-pass filter discussed in chapter 5.1.
The fk-diagram of a real P arrival recorded at YKA (13-jul-1994 10:49) is
displayed in Figure 5.2 d). The phase shows a slowness comparable to the
theoretical slowness. The theoretical slowness and backazimuth for P (u =
4.43 s/, = 295.5) and PP (u = 7.65 s/,
= 295.5) are marked by the white diamonds. The deviation
of the backazimuth from is within the azimuthal resolution
of the fk-analysis at YKA. The form of the fk-diagram is similar to the
ARF. This resemblance indicates a large coherency of the phase in the fk time
The additional information obtained by the fk-analysis is used to
identify coherent arrivals and to distinguish them from incoherent noise in
the PP study.
Next: Characterization of PP phases