Seismology Research
Bringing together seismology and atmospheric sciences
6. Processing techniques, frequency slowness analysis and Fisher statistics
January 2004
Läslo Evers
In this chapter a processing technique for infrasonic data is described. Data are processed
in the frequency domain on the basis of signal coherency. The Fisher value, a measure of
signal coherency, is used to detect a coherent infrasonic wave traveling over the array.
The technique is examplified with the recording of a sonic boom.
6.1 Theory
The frequency slowness power spectrum can be calculated in a signal summation, according
to
Smart and Flinn, 1971:
This formula can be interpreted as a shift of the array response by a infrasonic wave
traveling coherently over the array. The signal characteristics are transformed to the
frequency domain, through the Fourier transform, where time differences per instrument
are translated to phase shifts.
The frequency slowness power spectrum is evaluated for all possible slownesses in the
infrasonic domain and all relevant frequencies. A specific frequency and slowness vector
will correspond to the maximum of the power spectrum, enabling identification of the
source. The apparent sound velocity is resolved as the length of the p-vector
and the azimuth is determined by its angle with repect to the North:
The best beam can be constructed after having solved the apparent sound velocity and azimuth.
The best beam can be imagined, in the time domain, as the sum of the recordings after alignment
according to the apparent sound velocity and azimuth. A further increase in signal-to-noise
ratio is achieved, since incoherent energy will add up negatively in this stack.
Data can be processed in the above described way. Signal detection can be achieved (automatically)
by evaluating signal coherency, by means of Fisher statistics:
The amount of signal, present in the recording, is described by:
The total amount of energy is:
Thus, the value of Fisher statistics is a scaled measure of signal coherency. An increase
of the Fisher value is caused by a coherent wave traveling over the array. Incoherent
energy, noise, will result in low Fisher values.
6.2 Example
In figure 6.1, the recording of coherent infrasonic energy traveling over DIA, is shown.
The travel time differences per instrument enable a first guess of the source's azimuth.
Figure 6.1: Coherent infrasonic energy traveling over DIA. Note the time differences
per instrument, these are used in the frequency slowness analysis as phase shifts to
identify the source.
Instrument 7, 8 and 9 are the first to record the energy. These instruments are located
to west of the center of the array, therefore we estimate the source to be situated
somewhere west of the array. The travel time differences and amplitude of the signal
will be used in the frequency slowness analysis to make an exact calculation of the
source characteristics, based on signal coherency.
In figure 6.2, the result of frequency slowness analysis combined with Fisher statistics
of the recordings is given. The time axis is the same in the four frames. The lower three
frames show a resolved parameter (i.e. coherency, apparent sound velocity and azimuth) as
function of time and frequency. The top frame displays the best beam of DIA.
Figure 6.2: The result of frequency slowness analysis combined with Fisher statistics. In
the lower frame signal coherency is plotted as a function of frequency. Apparent sound velocity and
azimuth are plotted in the middle frames. The top frame shows the best beam of DIA, i.e.
the sum of the 16 recordings, aligned according to the resolved azimuth and apparent sound velocity.
Signal coherency, as Fisher value, increases around the signal's arrival (between 5 and
20 seconds), as follows from the lower frame. The signal is approximately coherent between 0.5
and 3.0 Hz, having its maximum coherency after 12 seconds at 1.0 Hz. The signal seems to have
traveled with a reasonable apparent sound velocity of approximately 350 m/s as identicated by
the green colors between 5 and 20 seconds in the lower middle frame. As expected, the signal appears
to have come from the West. The yellow colors for the azimuth plot (top middle frame) denote
an incoming angle of the enery between 270 and 360 degrees, thus around Northwest.
The exact values of apparent sound velocity and azimuth
are known in the calculations, creating the plots in figure 6.2. With the values at maximum
signal coherency, the best beam in the top frame of figure 6.2 was constructed. Figure 6.3
is displayed to illustrated the use of frequency slowness power plots for the derivation
of the source characteristics.
Figure 6.3: Frequency slowness power plot at 1.0 Hz of a time window around the event.
The white p-vector resolves the apparent sound velocity of 346 m/s by its lenght and the azimuth of
318 degrees as its angle with respect to the North.
As described, frequency slowness analysis can be interpreted as a shift of the array response
by an amount determined by the phase shifts per instrument of the infrasonic wave traveling over
the array. In figure 6.3, the array response (normally centered around (px,py)=(0,0))
is shifted towards the northwestern quadrant of the graph. The amount is given by the white
slowness vector (p-vector) and resolves an apparent sound velocity of 346 m/s by its lenght
and an azimuth of 318 degrees by its angle with respect to the North.
