IEEE Signal Processing - March 2018 - 56

within data or the repeated patterns in the
the time delays. However, as depicted in
As the density of seismic
sequence, from exploiting the more complex
Figure 3, in this case we would need longer
exploration surveys
or accurate statistical properties within the
filters to capture the similarities between
increases, the correlation signal. This leads to a significant degradation
traces to have effective predictions. On
among traces becomes
in performance unless the available signal
the other hand, longer filters have the dismore significant. Past
length is prohibitively long.
advantages of increased computational
To alleviate the problems encountered in
complexity and slower convergence speed
works have given limited
large
alphabet signal compression, we proof the adaptation algorithm, which in turn
attention to exploit the
pose
an
approach based on oversampling
decreases the compression performance.
interdependency of the
but a representation with fewer number of
Figure 4 compares the results of using
sensory data.
bits. In a typical A/D converter, sampling
two previous traces (in addition to the curabove Nyquist rate is sufficient for perfect
rent trace) in the RLS predictive coding
reconstruction. If the sampled signal is quantized with Q
method (hereafter referred to as multitrace RLS) with that
of the conventional RLS (hereafter denoted as single-trace
bits, the average SNR is proportional to fs 2 2Q, where fs is the
RLS), and the (single trace) linear predictive coding (LPC)
sampling frequency. Note that the average raw bit rate from A/D
method. In all methods, we used a uniform quantization of
is fs Q. To reduce the number of quantization levels, the samthe residual signal followed by six-level context tree weightpling frequency should be increased exponentially for a fixed
ing (CTW) compression [34]. As shown in Figure 4(a), there
reconstruction SNR, which causes an exponential growth in the
is a minor performance enhancement by using multitrace RLS
raw bit rate from A/D, making it infeasible in most applications.
algorithm on the University of Utah's database compared to a
Our key idea of tackling the large alphabet signal compression
simple RLS adaptive filter on single trace, which is due to the
is to create 1-bit samples from the signal. However, to achieve
fact that the traces are not ordered according to the proximity
such an objective, we would require a nonlinear reconstruction
of source shots. However, for the USGS data shown in Figmethod. As such, we modify and use a 1-bit quantization of sigure 4(b), a significant performance improvement is achieved
nal x (t), proposed in [35] and [36], which can be cast as
by multitrace RLS, especially at high compression gains. For
example, at a compression gain of 20, more than 5 dB increase
x [n] = sign (x (nx) - d (nx)),
(6)
in SNR is attained.
where d ($) is a periodic dither function (known at both the
encoder and decoder) and x is the sampling time interval,
Oversampling: Tackling the large alphabet problem
much
smaller than the Nyquist sampling period. Alternatively,
The dynamic range of the resulting residual signal in predicthis quantization scheme can be represented by the (quantive coding is often too high. Therefore, the alphabet size of the
tized) times, t k, that the signal x (t) crosses the dither d (t),
quantized residual signal may grow too large. The large alphabet characteristics of seismic signals prevents the compression
x (t k) . d (t k), resulting in a nonuniform signal-dependent
algorithms, which rely on learning either statistical dependencies
sampling. In what follows, we refer to the time instances t k

140

100

100

60

SNR (dB)

SNR (dB)

80

40

20

20
0

60

0

10

20
30
40
Compression Ratio

50

-20

60

0

10

20
30
40
Compression Ratio

(a)

(b)
Single-Trace RLS

LPC

Multi-Trace RLS

Figure 4. Using multiple traces on RLS performance: (a) uncorrelated traces and (b) correlated traces.
56

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