estimates. Wavenumber estimates based on seis-
5. PROCESSING RESULTS FOR
mic signals from the moving tracked vehicles
FIELD RECORDS
tended to be reliable at frequencies between 9 and
This section presents ML wavenumber esti-
15 Hz despite the occurrence of peak spectral
mates of Grayling field data for the .45 caliber
energies in the vicinity of 30 Hz. The 30-Hz bins
blank pistol, the sledgehammer, and the M60 tank
rarely produced satisfactory wavenumber esti-
moving at 4.5 m s1. All wavenumber estimates
mates.
were formed over a regular grid space containing
Several factors were useful in qualitatively de-
2601 points. All spectral estimates contain various
termining the accuracy of wavenumber spectra.
degrees of bias and variance. To demonstrate the
The most important was the presence of a strong
degree to which frequency domain bias and vari-
array factor. This was particularly relevant when
ance can affect wavenumber estimates, we show
assessing the quality of wavenumber estimates
several results that use different OBAFFT param-
based on small numbers of OBAFFT blocks. An-
eters on the same M60 signal vector.
other key indicator proved to be the estimate of the
phase velocity. An estimated phase velocity con-
sistent with the surface wave velocity found by the
5.1. Criteria used to select processed frequencies
Several criteria were used to select the frequency
time-domain move-out analysis was usually a good
indicator of an accurate wavenumber estimate. In
bin at which a wavenumber estimate is formed.
addition, if the spatial correlation is normalized
The most important of these processing criteria
according to eq 25 then, when using the Bartlett
was the average array coherence spectra, which is
method, a peak power value of 0.6 or greater often
formed by calculating and summing the magni-
tude-squared coherence spectra (Γave) between all
accompanied an accurate estimate.
possible channel combinations and then normal-
izing by the number of combinations. This opera-
5.2. Impulsive sources
The first series of wavenumber estimates is
tion is given as
based on the vertical-geophone subarray response
Xil( f ) Xj*l( f )
to the .45 caliber blank fired from the CPA at θ =
^
^
1 M
∑
40. The waveforms of this signal vector are shown
^ l
^l
j=i+1 X ( f ) X ( f )
i
j
in Figure 17. Figure 26 shows the average array
amplitude and coherence spectra, based on a 2000-
nblks M1
2
^
Γave =
∑∑
point series with three blocks. Each block contains
(33)
(
)
(nblks) M2 M l1 i1
1024 points, is overlapped by 53%, and is tapered
using a Blackman window
Recall from section 4 that there are two distinct
In general, wavenumber spectra which were
arrivals in this record. The wavenumber estimate
processed at frequency bins with coherences
shown in Figure 27 is the Capon maximum-likeli-
greater than 0.65 and a coherence variance across
hood estimate processed at 78 Hz. This frequency
immediately adjacent bins of less than 0.025, pro-
was chosen since it produces a wavelength com-
duced reasonable wavenumber estimates. How-
parable to the spacing of the center elements of the
ever, it was found that wavenumber estimates
geophone array with an acoustic propagation ve-
could still vary by a considerable degree even with
locity. The time-domain interval considered was
average coherences of 0.9 (usually this was accom-
between 0.1 and 0.25 s. This corresponds to a time
panied by high coherence variance across adjacent
window that brackets the initial impulsive arrival
bins). Such variation can be explained by recogniz-
seen in Figure 17. There were 230 sample points in
ing that the coherence spectrum is also susceptible
this interval. The estimate was formed using a
to bias and variance errors.
block length of 128 points with each block over-
The average array coherence spectrum is
lapped 60%, giving a total of three blocks. Boxcar
strongly affected by the OBAFFT parameters. The
windowing was applied. The coherence at 78 Hz
Blackman taper normally produced a smoothly
varying coherence spectrum with peaks ranging
was 0.7. The wavenumber spectrum (Fig. 27) pro-
duces an estimated source direction of 40 and a
over several frequency bins. In contrast, a boxcar
phase velocity of 338 m s1. This is consistent with
taper often produced an erratic coherence spec-
trum. In general, peaks in the coherence spectra
the direction of the pistol shot and the acoustic
greater than 0.7, which varied smoothly through
propagation velocity. Note that the convention
several bins, produced acceptable wavenumber
followed plots the true vector orientation of the
22
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