Multiresolution analysis (MRA) was formulated based on the study of orthonormal, compactly supported wavelet bases. Wavelets theory and its applications are rapidly developing fields in applied mathematics and signal analysis. Wavelet basis representation of certain signals show advantages over the traditional Fourier basis representation both theoretically and practically. The MRA concept was initiated by Meyer [12] and Mallat [11], which provides a natural framework for the understanding of wavelet bases. Here, I give a brief description of orthonormal, compactly supported wavelet bases; detailed information can be found, for example, in Daubechies [6] and Jawerth and Sweldens [10].
An orthonormal, compactly supported wavelet basis of 
is formed by the dilation and translation of a single
function 
, called the wavelet function:
where 
is the set of integers.
In equation (1), the function 
has 
vanishing
moments up to order 
, and it satisfies the following
``two-scale'' difference equation,
The wavelet function has a companion,
the scaling function 
, which also forms a set of orthonormal
bases of ,
The scaling function satisfies,
and the ``two-scale difference'' equation,
In equations (2) and (5), two coefficient sets
and 
have the same finite length 
for a certain
basis, where is related to the number of vanishing moments in
. For example, equals 
in the Daubechies wavelets.
In the wavelet representation of signals, 
behaves as a low-pass filter and 
behaves as a
high-pass filter to signals. These two filters are related by
and are called quadrature mirror filters (QMF). An extensive study of the QMF can be found in [13].
: Illustration of the sequence of multiresolution analysis
subspaces 
. 
is the orthogonal complement of in
. Space 
represents the space that contains the finest
resolution data, and 
.
The MRA of is a set of nested, closed subspaces
, such that
where the basis for the subspace is a set of orthonormal,
translated functions, and each of these functions sets is a fixed
dilation of the scaling function, 
.
Therefore, these subspaces have the property
Defining to be the orthogonal complement of in ,
they are related by
The wavelet basis 
, as in
equations (1) and
(2), forms the orthonormal basis of the subspace .
Therefore, for 
, we can have
Figure 1 illustrates the nesting of subspaces and
their orthogonal complements .
In Figure 1, contains the original data which has
the finest resolution; the projection of the data on 
has increasingly coarser resolution.
In this paper, the data projected onto the subspace is referred
as the decomposition of data at resolution level 
.
We define the projection of a function 
on to be
. Then the th resolution level of the function has the form
where 
is the projection of the function 
on the basis
; that is,
Next, define the projection of on the subspace to be
where 
is the projection of function on the basis
Then, equation (10) implies that the original
function 
can be represented by
:
Decomposition of a Ricker wavelet at increasingly coarser resolution
levels. The bases of the decompositions are Daubechies wavelets with
and 
vanishing moments for the left and right figure
respectively. The first traces represents the signal at the finest
level, which is the original signal.
Figure 2 shows the decomposition of a simple
synthetic seismic trace at various resolution levels for two different
wavelet functions. The original trace is a Ricker wavelet,
i.e. a normalized second-order derivative of a
Gaussian function, with a peak frequency of 
Hz. The left figure
shows the decomposition by a Daubechies orthonormal basis with
vanishing moments, while the right figure shows the same decomposition
with vanishing moments. The Ricker wavelet (
) is the left
most trace in each box, while the remaining traces correspond to
of equation (11), where 
, respectively.
From Figure 2, it can be seen that the decomposed
traces contains progressively lower frequencies with the increase of
decomposition levels while the major features of the original signal
are preserved. Comparing the two plots in Figure 2,
we also observe that the increasing the number of vanishing moments
increases the smoothness of the decomposed signal.
