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

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, :

(1) |

(2) |

The output of the array can be computed by

(3) |

(4) |

(5) |

using Parseval's theorem. In equation (5.5), S) is the Fourier transformation of and

(6) |

and the wavenumber vector for . The backazimuth determines the direction of and the slowness the magnitude of . Equation (5.5) can be written as:

(7) |

(8) |

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 vector

(9) |

The backazimuth can be calculated by

(10) |

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 is normalized

(11) |

(12) |

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
window.
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.