In many applications, it is required that the processes
applied to the obtained signals be shift-invariant. For example, in
examining the multi-scale property in residual statics correction
problems, it is important that the error-fitting function at each
scale have a common - or at least close to common - global
minimum.
Therefore, we expect that the relative time-shifts among traces at
each scale to be almost the same as it was in the original data, and
that the waveforms not be deformed from one trace to another.
However, the orthonormal wavelet bases representations are generally
not shift-invariant.
This shift-variance can be seen directly from the construction of
their bases, equations (2) and (5), because of
the change of step sizes among different scales in these definitions.
Therefore, the Daubechies wavelet bases are not suitable for our
purpose.
Figure 3 shows ten copies of randomly shifted
Ricker-wavelet traces, and their projections onto the subspace 
in the Daubechies bases with 
vanishing moments.
The decomposed waveforms on the right of Figure 3
are deformed to different shapes among traces with different
time-shifts, and they do not have the same relative time shifts of
those shown on the left of Figure 3.
:
Ten traces of randomly shifted Ricker-wavelet traces (left) and their
decomposition at resolution level 
in the Daubechies wavelet bases
with vanishing moments of (right).
:
The decomposition of ten copies of randomly shifted Ricker-wavelet
traces, in the shell of the Daubechies basis with vanishing
moments, at resolution levels and 
. The original traces are
shown on the left of Figure .
:
The decomposition of ten copies of randomly shifted Ricker-wavelet
traces, in the auto-correlation shell of Daubechies basis with
vanishing moments, at resolution levels and . The original
traces are shown on the left of Figure .
Saito and Beylkin [19] suggested using the shell
of an orthonormal basis when shift-invariant is required.
Without loss of generality, let us assume that the signal we consider
having finite length 
. Consider a family of functions
where
where the functions 
and 
are a wavelet and scaling
function, respectively. The new family of functions defined by
equations (14 and (15 can also serve
as bases for subspaces 
and 
in MRA. They are complete, but
they are redundant and not orthonormal [18].
Therefore, the decomposition of a function in these bases is not
unique. However, by forcing an additional constraint to the projection,
a function 
may still be decomposed in the shell of an
orthonormal basis much the same way as it was in an orthonormal
wavelet basis itself. In this case, the basis functions in
equations (11) and (12) are
replaced by 
and 
.
The representation of signals using this family of bases are
shift-invariant among different scales. Figure 4
shows the same numerical experiment as that in
Figure 3, except using the shell of orthonormal
bases expansion at resolution levels and .
The relative time-shifts among traces are preserved while the
waveforms are deformed to the same amount.
However, the original symmetric waveforms are deformed to asymmetric
waveforms. This deformation of the waveforms is not desirable, and may
cause problems for some applications.
To overcome this problem, a family of symmetric, shift-invariant bases
are introduced [19].
Let 
and 
be auto-correlation functions of scaling
function and wavelet function respectively,
where 
and 
satisfy equations (2) and
(5) respectively.
Construct a family of bases
where
Now, we have an auto-correlation shell of an orthonormal basis
that is both symmetric and shift-invariant. Figure 5
shows the expansion of shifted Ricker-wavelet traces in the
auto-correlation shell of Daubechies basis.
It can be seen that both the symmetry of the waveforms and the
relative time-shifts are preserved at resolution levels and .
There exists a fast algorithm for expanding a function
using the auto-correlation shell of orthonormal basis [19].
I only give the formulas of the discrete expansion;
detailed derivation can be found in [19].
Suppose that 
and 
are the projected signal onto the
subspaces and at the sampled positions respectively,
that is
where 
is the sampling interval.
Then, two symmetric filters, 
and
are applied recursively to the
signal we wish to decompose,
where 
, 
, and 
is the filter length
in the ``two-scale difference'' equations of wavelet and scaling
functions as in equations (2) and (5).
In equation (20), is the number of samples of the
signal and the filter coefficients 
and 
are,
and
In equations (21) and (22), coefficients
are the correlation of the low-pass filter
in equation (5),
