IEEE Signal Processing - March 2018 - 59

discrete stationary source X with probability distribution
P (x), it is known that the minimum code length is lower bonded by L = E {log 2 (P (X ))}. We may estimate the probability
distribution of a sequence by first assuming a parametric family of distributions, and then computing the parameters of the
model from observed data. As such, we can obtain a bound on
the minimum code length. As the model fits the data better,
this bound becomes tighter.
To apply the aforementioned mechanism, we first note
that the seismic traces are nonstationary and correlated. Thus,
we consider the residuals of predictive coding filters for the
distribution modeling. Since the data from neighboring sensors are highly correlated, the residuals of all available shots
from such sensors are stacked to create a vectorized data set
on which we apply a noisy independent component analysis
(ICA) for modeling. That is, x (i) = As (i) + e (i), where x (i) is a
vector of length n consisted of residual values at the ith data
point from n neighboring sensors, A is the mixing matrix, s (i)
is the generative signal of dimension m, and e (i) is assumed
to be multivariate Gaussian noise. This method is inspired by
the seismic convolutional model, which is used in its deconvolution and reflection analysis. This model approximates the
earth as a linear system. In this linear model the seismic trace
is assumed to be generated by the convolution of a wavelet (i.e.,
the source shot) by an unknown filter corresponding to the
reflection coefficients of the earth's layers, and contaminated
by noise. As in ICA, the components of the generating signal
s = [s 1, f, s m] T are assumed to be independent and have nonGaussian distributions. For simplicity, we adopt a GMM for
probability estimation of the generating signals, i.e., s i is a mixture of K i Gaussian sources, each with weight w ij and distribution N (n ij, v 2ij), j = 1, f, K i . Therefore, the resulting average
code length is equal to L = ^1/nN h R iN= 1 - log P (x (i) | H),
where H is the set of all model parameters including GMM
parameters for each source, mixing matrix A, and noise covariance R. Note that P (x | H) = # P (x s, H) P (s) ds, where
x s, H + N (As, R) is multivariate Gaussian. Further, assuming independence of the sources, we get P (s) = P mi = 1 P (s i),
where P (s i) = R Kj =i 1 w ij N (n ij, v 2ij). We refer to the optimization problem with the average code length computed from
the aforementioned data generation model as ICA-iGMM (in
which iGMM stands for an individual GMM model for each
source). Because of the intractability of the integral in the
previously given equation, we used variational Bayes (VB) to
solve the problem [38], [39].
Alternatively, we can avoid variational methods and instead arrive at a closed-form solution for the likelihood
function, P (x | H). To achieve this goal, we model the
joint distribution of s is as a multivariate Gaussian mixture
P (s) = R Kl = 1 c l N (n l, K l), where K = P i K i and K l s are the
diagonal covariance matrices in the mixture. It can be shown
that P (x H) = R Kl = 1 c l N (An l, R + AK l A T ), where c ls, n l s,
and K l s can be obtained from the individual GMM parameters
of s i s. We refer to the associated optimization problem as ICAjGMM (i.e., joint GMM for sources). We used the GD method
to solve the resulting optimization problem. Note that the previ-

Table 2. Results of the average code length obtained via various
schemes on seismic data.
Method

USGS Data

University of
Utah Data

RLS residual + CTW

19.0

12.9

Multivariate Gaussian

17.4

13.0

ICA-jGMM + GD

16.1

12.2

ICA-iGMM + VB

16.2

12.7

Table 3. Lossless compression ratio for RAR, CTWL, and CTWL+clustering.
Method

Compression Ratio

ZIP (LZ)

1.76

7Z (LZMA)

1.90

RAR

2.07

CTWL

2.13

CTWL + five clusters

2.95

ously given approach is prone to overfitting. By increasing the
number of sources and their parameters, we may get a higher
fitting likelihood. However, we avoided overfitting by penalizing the objective function through incorporating the model
complexity (i.e., the bit overhead to represent the model).
Table 2 shows the resulting average code lengths for both
proposed estimation methods ICA-iGMM and ICA-jGMM.
As shown in the table, there is a noticeable gap between the
average code length achieved by the state-of-the-art lossless
compression scheme (i.e., RLS+CTW) and the bound on the
minimum code length estimated by our model. This result suggests that there is still room for the improvement of the lossless compression schemes. The table also indicates that the VB
method works as well as the more complex GD approach. The
results also suggest that over the USGS data set, the Gaussian
approximation of the residuals is not too accurate and, as evident from the table, the proposed models lead to better results.

Discussion and future work
In this article, we presented a summary of different approaches
for lossy compression of seismic signals. Some of the recent
advances, such as rate-optimized DL, multitrace RLS, and
oversampling methods, have demonstrated significant improvements in compression gain in various SNR regimes.
The existing methods mostly rely on mean squared error
to measure the quality. Although this measure achieves the
desired signal quality on average, it is unclear how the loss is
distributed over different parts of the signal and how it affects
the final product at the later stage (e.g., the seismic interpretation stage). One possible future research direction is to properly distribute the compression loss over the seismic trace to
achieve a higher compression gain without sacrificing the
performance of the seismic interpretation stage. For example,

IEEE Signal Processing Magazine

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March 2018

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