Note: Descriptions are shown in the official language in which they were submitted.
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Method of using matrix rank reduction to remove random noise from seismic data
processed in the f-xy domain
FIELD OF THE INVENTION
The resent invention relates generally to processing seismic data and
particularly to
P
reducing noise in seismic data using a variety of 3D eigen filtering
techniques based on matrix
rank reduction.
BACKGROUND OF THE INVENTION
Seismic data can be used to interpret or to infer sub-surface geology, making
it useful for
the location, identification and exploitation of petroleum and minerals.
However since seismic
traces are often contaminated by random noise, seismic data must undergo a
series of
statistical processes (known as "seismic processing") before it can be so
used. It is
advantageous to remove noise at as early a stage in processing as possible,
since this
improves the ability to perform all subsequent processing work. Such
conventional statistical
processes disadvantageously degrade or distort the required signal without
removing sufficient
noise. Conventional statistical processes are based on functions and their
transforms that it is
often useful to think of as occupying two domains. These domains have been
referred to as the
function and transform domains, but more commonly they are known as the time
and fre4uency
domains. Operations performed in one domain have corresponding operations in
the other. For
example, the convolution operation in the time domain becomes a multiplication
operation in the
frequency domain and the reverse is also true, permitting users to move easily
between
domains so that operations can be performed where they are easiest. The
seismic data
processing industry traditionally operates in the time domain.
Prior Art Figure 1 illustrates a typical geological exploration setup for
acquiring seismic
data. Positioned at or just below the earth°s surface 110, an energy
source 120 (typically
explosive high-energy or vibrational low energy) generates at least one sound
wave (a short
e.g. 3 sec. duration acoustic impulse or a longer e.q. 4-10 sec. duration
sweep) or signal having
sufficient energy to follow path 130 down into the earth a suitable distance
to reflect at the
interface of any changes in geology (commonly known as "events" or
"reflectors") 140, the
reflected energy traveling back to the surface via path 150 is simultaneously
recorded by
receivers 160 (commonly geophones positioned as an array for 3D, or a line for
2D exploration).
In marine seismic the sound wave is generated just below the surface of the
water and the
reflected energy detected by hydrophones. For each such sound wave generation,
or "shot", the
reflected signal returning to the surface via path 150 creates a "trace",
which is a single
recording at a receiver 160. Traces are detected and recorded in the form of a
time series of
sample measurements of particle velocity (for land data) or pressure change
(for marine data).
Many shots are taken to generate a seismic data set, often resulting in
hundreds of millions of
traces that may be stacked of summed in a variety of ways. When high energy
impulsive
seismic sources are used, the level of the detected true earth response
seismic signal is usually
greater than the ambient noise. However, when low energy surface seismic
sources are used,
the ambient noise can be at a level greater than the true earth response
seismic signal. For this
reason, seismic-trace recordings are often made involving the repeated
initiation of a low energy
surface seismic source at about the same origination point, thereby producing
a sequence of
seismic-trace data based on seismic wave reflections and/or refractions that
have traveled over
about the same path and therefore have approximately the same travel times.
The process of
adding such seismic-trace data together far improving the signal-to-noise
ratio of the composite
CA 02402942 2002-09-12
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seismic-trace recording is known as "vertical compositing" or "vertical
stacking." It should be
distinguished from "horizontal stacking," a process applied to a sequence of
seismic-trace data
based on seismic wave reflections from approximately the same subsurface point
(referred to as
the "common-depth point," or "CDP") but which has been generated and recorded
at different
surface locations. Horizontal stacking of CDP seismic-trace data requires that
time corrections
(called "normal moveout," or "NMO," corrections) be applied before the traces
are summed
together, since travel times from seismic source to detector are unequal for
each trace in the
sequence. It can be assumed that the true earth response seismic signal
embedded in each
trace is coherent and in phase (correlated} and that the noise is random and
incoherent
(uncorrelated) with zero mean value. In general, the objective of vertical
stacking is to maximize
the signal-to-noise ratio of the resultant recording.
Reflectors that are not "flat" are said to "dip" or slope.
The use of a low energy vibrator can be more economical than the use of
dynamite.
Furthermore, as compared to the use of a high-energy impulsive seismic source,
such as
dynamite, the frequency of the seismic waves generated by a vibrator can be
selected by
controlling the frequency of the pilot signal to the power source, such as a
hydraulic motor,
which drives the vibrator. More particularly, the frequency of the pilot
signal to the vibrator power
source can be varied, that is, "swept," for obtaining seismic-trace data at
different frequencies. A
low energy seismic wave, such as generated by a vibrator, can be used
effectively for seismic
prospecting if the frequency of the vibrator "chirp" :signal which generates
the seismic wave is
swept according to a known pilot signal and the detected seismic wave
reflections and/or
refractions are cross-correlated with the pilot signal in order to produce
seismic-trace recordings
similar to those which would have been produced with a high energy impulsive
seismic source.
Typically, the pilot signal is a swept frequency sine wave that causes the
vibrator power source
to drive the vibrator for coupling a swept sine wave "chirp" signal into the
earth. The swept
frequency operation yields seismic-trace data that enables different earth
responses to be
analyzed, providing a basis on which to define the structure, such as the
depth and thickness, of
the subsurface formations. It is a problem that recorded seismic-trace data
always includes
some background (ambient) noise in addition to the detected seismic waves
reflected and/or
refracted from the subsurface formations (referred to as the °'true
earth response"). Noise can
be classified as "stationary" and "non-stationary", both of which can be
random. Stationary noise
is random noise such as atmospheric electromagnetic disturbances that are
statistically time-
invariant over the period of acquisition of seismic-trace data for producing a
recording. Non-
stationary noise is random and often occurs as bursts or spikes generally
caused by wind,
traffic, recorder electrical noise, et cetera, which are statistically time-
variant over the period of
acquisition of seismic-trace data for producing a recording and exhibits
relatively large
excursions in amplitude. In connection with swept frequency operation of low
energy vibrator
seismic prospecting, it is common practice to vertically stack, or sum, the
seismic-trace data
from a series of initiations, that is, sequential swept frequency operations,
to produce a
composite seismic-trace recording for the purpose of improving the signal-to-
noise ratio of the
seismic-trace data. Unfortunately, the commonly used technique of vertically
stacking trace data
is inadequate in the presence of non-stationary noise that appears during such
seismic
prospecting.
Seismic data is acquired in two principal geometries: 2-D and 3-D. In 2-D
acquisition,
shots and receivers are positioned along a (not necessarily straight) surface
line. In 3-D
acquisition, shots and receivers are positioned over a 2-D surface area.
Seismic data related to
3D geologic volumes necessarily includes random noise that may be isolated
from the signal
data to different degrees by different conventional techniques, including an
eigenimage filtering
CA 02402942 2002-09-12
technique that is 2D in nature and disadvantageously does not account for
additional available
information respecting the formation.
For 2-D acquisition the main product of seismic processing is a 2-D stacked
"section"
(illustrated in Prior Art Figure 2), one of the dimensions representing
horizontal position along
acquisition line 210, and the other dimension representing time 220. For 3-D
acquisition,
seismic processing resulting in a 3-D stacked section (illustrated in Prior
Art Figure 3), two of the
dimensions representing edges 310 and 320 of the acquisition surface area, and
the other
dimension representing time 330.
Known seismic processing arrangements include common-midpoint (CMP) stacking,
where traces are collected into groups having roughly the same midpoints
between the
locations of the shot by which they were generated and the receiver at which
they were
detected. For each recorded time sample, the magnitudes or values of every
trace in the group
are summed together, producing a single "stacked" trace for each group. Such
stacking
commonly reduces the amount of data that must be processed by a factor of
between 10 and
100.
Geological interpretation is easiest and most successful on seismic sections
having low
levels of noise, and thus one of the objects of seismic processing is to
remove as much noise as
possible. Noise can be broadly categorised as random, coherent, or
monochromatic. Random
noise may be defined as noise that is uncorrelated between traces and
spectrally broad band.
Some of the causes of random noise are the effects of wind and other
disruptions on the
seismic receiver and cable, poor penetration of seismic energy through the
earth (particularly in
the near surface beneath the shot or receiver), and numerous natural and man-
made seismic
energy sources apart from the intended one. The most common stage to carry out
random-
noise removal is after CMP stacking. A number of methods have been developed
to do this,
including: f-k transform (March and Bailey, 1983), f-x prediction (Canales,
1984; Soubaras,
1994), Karhunen-Loeve transform (Jones and Levy, 1987; AI-Yahya, 1991 ),
eigenimage
(Ulrych, Sacchi, and Freire, 1999), spectral matrix filtering (Gounon, Marse,
and Goncalves,
1998), and Radon transform (Russell, Hampson, Chun, 1990(a) and 1990(b)).
The foregoing methods work on 2-D data sets, but can be adapted for 3-D
stacked
sections by slicing the data volume along one of them spatial dimensions,
filtering each of these
slices separately as if it were a 2-D section, and then recomposing the 3-D
volume. This can
then be repeated in the opposite spatial direction. Such methods are not
optimum in that they
fail to fully exploit the large amount of data available within a short radius
of each spatial point of
the 3-D volume. For this reason, "true 3-D" methods have been developed that
work in both
spatial dimensions simultaneously, including: f-xy prediction (Chase, 1992;
Soubaras, 2000)
and f-kk transform (Peardon and Bacon, 1992).
Random noise removal before CMP stacking is less common. There are, however,
at
least two advantages to removing random noise as early in the processing
stream as possible.
First, it improves the performance of subsequent processes, notably
deconvolution, statics
correction, and velocity analysis. Second, it has the potential to be more
effective since more
data is available before stacking, providing better statistical redundancy. At
the same time, extra
data means that random noise removal before CMP stacking requires more
computation. Noise
removal before statics or deconvolution faces the problem of "surface-
consistent effects",
meaning effects that are constant within each shot and receiver, but that may
change radically
even between adjacent shots and receivers. If these effects have not been
corrected before
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noise removal then the noise removal process must preserve them, one method
for this is
surface-consistent f-x prediction (Wang, 1996).
Another application of noise removal is common-offset or common-angle stacks
for
amplitude versus offset (AVO) or amplitude versus angle (AVA) analysis. Such
stacks are used
for the automatic computation of parameters for interpretation. These stacks
require a low level
of noise so the computed parameters are as accurate as possible. In 2-D
acquisition, AVO/AVA
stacks form a 3-D volume in which the two spatial dimensions are CMP and
either amplitude
versus offset or amplitude versus angle. In the offset/angle dimension there
may be only one or
two dozen traces, such that much of the data is on or near a spatial boundary.
Disadvantageously even the better noise removal methods, such as f-xy
prediction, do not
perform well near spatial boundaries, resulting in distortion of the signal,
and possible distortion
of the computed parameters - creating the need for a noise removal method that
performs well
at spatial boundaries.
Some conventional noise removal methods such as Karhunen-Loeve, time-domain
eigenimage, Radon transform, and f-k and f-kk transforms, only work well on
plains-type data
received from geological formations in which most of the reflectors are flat.
Disadvantageously
on more structured data received from geological formations in which
reflectors are strongly
dipping (also known as sloping), these methods become either computationally
expensive,
difficult to use, or less effective. There is a need for a method of noise
removal that performs
well on all types of geology. Disadvantageously, time-domain eigenimage
filtering is not well
suited for structured data.
Many methods of noise removal from seismic data are implemented using matrix
operations.
Matrix compression is the process of determining a representation of a given
matrix, or
a representation of an approximation to a matrix, which representation
consumes using less
space than said given matrix itself. However, matrix rank reduction is to
determine the nearest
(with respect to a particular matrix norm) rank-k matrix to a given matrix. A
matrix norm
measures the size of a matrix, for example, the "Frobenius norm" is the square
root of the sum
of the square of the matrix elements, whereas the ''L1 matrix norm" is the sum
of the absolute
values of the elements of the given matrix.
Prior art reviewed includes US 5,379,268 to Hutson, who teaches compression of
the
subject matrix (see claim 1 (b)) as the core of his improvement. Compression
and rank reduction
have some similarities, but are not identical, in that rank reduction can be
used as a step within
matrix compression, but rank reduction can also be performed without any
resulting matrix
compression. Hutson in 268 teaches compression - by active decomposition
(actually
expanding the original data matrix), then actively and selectively zeroing
(and it is important that
modifying to a value other than zero does not "compress") singular values in
one of the resulting
matrices, thereafter recomposing the matrices into a single matrix that is
representative of the
original - after which 268 proceeds to teach (see claim 1 (c)(1 )) further
processing of the signals
represented by the compressed matrix. However, 268 is vague even about the
kind of
compression and processing performed. For example modifying without zeroing
cannot
compress while zeroing inappropriate selections can actually be
counterproductive by
eliminating signal rather than noise. Disadvantageously, existing noise
removal algorithms do
not handle erratic noise well. For example, both SVD and Lanczos methods of
matrix rank
reduction attempt to find a rank-k matrix that is nearest to the input matrix
in a Frobenius-norm
sense. This is appropriate for removing random noise that has a Gaussian, or
bell-shaped,
CA 02402942 2002-09-12
statistical distribution - which does not perform as well when erratic non-
Gaussian noise bursts
are present.
Processing seismic data is time consuming and expensive because it involves
large
quantities of complex data. Therefore, it is desirable to provide a solution
to at least some of the
above-described problems of the prior art reducing either or both the quantity
of data processed
or the amount of processing required in relation to that data. The prior art
in the seismic data
processing industry has concentrated on teaching variations on: time domain
based and fully
decomposed matrix operations, and none of the prior art reviewed is based on
matrix rank
reduction.
SUMMARY OF THE INVENTION
In accordance with one of its aspects the present invention comprises a method
based
on finding the rank k matrix nearest to a given matrix after which the
smallest k value is used
based on which the difference plot show insignificant signs of signal.
According to the method
aspect of the present invention random noise is removed from a seismic data
set by the
following steps: the subject seismic data set is first divided spatially into
many small,
overlapping, rectangular grids of traces; each rectangular grid of traces is
processed
independently by first transforming the grid traces into the frequency domain
using a Discrete
Fourier Transform (DFT); the grid is then separated into constant-frequency
slices; all or a
subset of the constant-frequency slices are then individually rank reduced; an
inverse DFT is
performed on each grid trace; and when the required grids are processed, the
subject seismic
data set is reformed from the processed rectangular grids. In the frequency x-
y plane for a given
frequency a rank K matrix is produced using eigen analysis wherein K is the
number of plane
waves, which fact allows the separation of plane waves by eigenimage
decomposition. A single
frequency slice is rank reduced by placing this 2-D grid of complex DFT values
into a complex-
valued matrix of the same dimensions, finding the nearest rank-k matrix to
this matrix (or an
approximation to this), where k is some value greater than or equal to one,
and replacing the
constant-frequency slice values with the values from the rank-k matrix.
In accordance with a method aspect of the present invention there is provided
a novel
statistical process for removing noise from seismic data sets, improving the
interpretability of the
final result. This novel method for removing noise works at almost any stage
of signal
processing. For example, since it preserves surface-consistent effects, this
method may be
applied before statics correction or deconvolution. It may also be used to
remove noise on 3-D
volumes of stacked traces as well as common-offset or common-angle stacks. In
an alternative
embodiment, this method can even be used to remove coherent noise from seismic
traces. For
stacked 3-D volumes this method can be executed faster than f-xy prediction
filtering. The
method of the present invention provides better results along the boundary of
the subject
volume. Good performance along the boundary and the ability to address non-
uniform shooting
patterns are further advantages.
In order to overcome the disadvantages of the prior art in one of its aspects
the present
invention comprises a novel method for removing noise from seismic data sets,
which method is
frequency domain based and less time consuming and expensive by reducing both
the quantity
of data processed and the amount of processing rE:quired in relation to that
data. According to
the method of the present invention based on matrix rank reduction it is not
necessary to fully
decompose the subject matrix or to take the active atep of zeroing out select
elements because
the partial decomposition results in the desired matrix elements being
extracted early in the
decomposition process, permitting decomposition to be terminated before
completion.
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(j
In accordance with one of its aspects the method of the present invention
commencing
with an m-by-n 2-D grid of seismic traces (The spatial locations of these
traces need not be
rectangular - they can, for example, form a parallelogram. And the distances
between grid lines
in either the row or column directions need not be evenly spaced) comprises
the steps: take the
Discrete Fourier Transform (DFT) of each trace in the grid; and for each
frequency in the
resulting DFT:
form an m-by-n complex-valued matrix A whose elements are the DFT values of
each
trace in the grid for the current frequency;
calculate an m-by-n rank k matrix approximation to A, where 1 <= k < min(m,n)
for the
purpose of creating a matrix B;
replace the trace DFT values for this frequency with the elements from said
matrix B;
repeating the foregoing process for an appropriate subset of all available
frequencies; and
thereafter taking the inverse DFT of each trace in the subject grid. The
amount of noise
removed by the foregoing method can be increased by increasing the grid
dimensions m and n,
or by decreasing the rank k. The 2-D grid of seismic traces may, for example
but not in
limitation, originate from:
A rectangle of traces extracted from a stacked 3-D volume. The trace grid
being
comprised of inline CDP bins in the row direction, and crossline CDP bins in
the column
direction;
Traces from an unstacked 2-D line. The grid is composed of common source
traces in
the row direction, and common receiver traces in the column direction;
Traces from an unstacked 3-D volume, where the traces are taken from a single
shot
line and receiver line. The trace grid being comprised of common source traces
in the
row direction, and common receiver traces in the column direction; or
Traces from common-offset or common-angle stacks for a sequence of CMPs. The
trace
grid being comprised of common-offset or -angle traces in the row direction,
and CMP
traces in the column direction.
According to one aspect of the present invention there is provided a
substantial
advantage for removing noise on common-offset or common-angle stacks, and for
removing
noise on pre-stack data, the present invention is superior to standard f-xy
prediction filtering
since it is faster, can preserve surface-consistent effects (allowing it to be
applied before statics
correction and possibly deconvolution).
An audio signal is being sampled at 8Hz, which means that at each successive
eighth of
a second a measurement of the intensity of the signal is taken. The Fourier
transform
decomposes or separates a waveform or function into sinusoids of different
frequency which
sum to the original waveform. It identifies or distinguishes the different
frequency sinusoids and
their respective amplitudes. The Discrete Fourier Transform (DFT) is required
because a digital
computer works only with discrete data, numerical computation of the Fourier
transform of f(t)
requires discrete sample values of f(t), which we will call fk. In addition, a
computer can compute
the transform F(s) only at discrete values of s, that is, it can only provide
discrete samples of the
transform, Fr.
The accompanying drawings, which are incorporated in and constitute a part of
this
specification, illustrate preferred embodiments of the method, system, and
apparatus according
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7
to the invention and, together with the description, serve to explain the
principles of the
invention.
BRIEF DESCRIPTION OF THE DRAWINGS
The present invention, in order to be easily understood and practised, is set
out in the
following non-limiting examples shown in the accompanying drawings, in which:
FIG 1. is Prior Art and an illustration of the typical system used in the
acquisition of a
single seismic shot.
FIG 2. is Prior Art and an illustration of a typical 2-D stacked seismic
section.
FIG 3. is Prior Art and an illustration of a typical 3-D stacked seismic
section.
FIG 4. is an illustration of a flow chart of one embodiment of one aspect of
the present
invention.
FIG 5. is a flow chart relating to removal of random noise in an individual
rectangular grid
of traces.
FIG 6. is a flow chart relating to reducing the rank of a matrix.
FIG 7. is a flow chart relating to the selection of rank k.
FIG 8. is a surface stacking diagram describing the acquisition of a 2-D
acquisition
seismic line.
FIG 9. illustrates the positions of shots and receivers in a 3-D acquisition
seismic array,
and the selection of a particular shot and receiver line for individual noise
removal.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
Reference is to be had to Figures 4 - 9 in which identical reference numbers
identify
similar components.
According to one embodiment of the method of the present invention Figure 4
illustrates
the removal of random noise from a stacked 3-D section of seismic data 410.
Said section 410
is spatially divided into at least 2 overlapping rectangular grids of traces
420. If a grid is missing
traces (usually because it is located near the spatial boundary of said 3-D
section), then insert
artificial traces with sample values of zero to complete it. Figure 5
illustrates the sub-steps of
step 430 necessary to remove random noise from each rectangular grid 430. For
each grid of
traces 510 transform each trace in the grid into the frequency domain using a
Discrete Fourier
Transform (DFT) 520, then separate the frequency domain grid into constant-
frequency slices
530. Each frequency slice 530 is placed into a complex-valued matrix of the
same dimensions
and the matrix is reduced 550 to a rank k, where k is some value greater than
or equal to one.
When the matrices of all frequency slices of said frequency domain grid are so
reduced, reform
560 the subject rectangular grid from the rank.-reduced matrices. The
rectangular grid may be
CA 02402942 2002-09-12
transformed back into the time domain 570 by taking the inverse DFT of each
trace, resulting in
a grid that is representative of the original grid but having a better signal
to noise ratio. Once all
rectangular grids are so noise-suppressed, reform the stacked 3-D section
using all said grids.
Overlap zones are addressed by summing grid traces at the same position after
scaling them
with weights so that the sum of the weights at the overlap position is one. A
person of skill in
seismic processing would know to select said weights to taper smoothly from
the boundary of
each rectangular grid.
There are a number of ways to rank reduce a matrix as required by step 550,
however,
Figure 6 illustrates the classical way to reduce the rank of a matrix, by:
applying Singular Value
Decomposition or SVD 620 to the subject matrix 610 to expand matrix 610 into
left singular
vectors, singular values, and right singular vectors - resulting in 3 separate
matrices, including
an ordered diagonal matrix. Classical reduction results at 630 when all but
the k largest singular
values of the diagonal matrix are zeroed, after which the matrix may be
recomposed 640 in rank
reduced form 650. The amount of noise removed is controlled by adjusting the
rank k and the
size of the extracted 2-D grids of traces {i.e. the row and column dimensions
of matrix 610).
Since there is a trade-off between these parameters, a good strategy is to use
the same size of
grid (e.g. 20 by 20) and adjust the rank to match the data set. The smaller
the rank, the more
noise is removed, but the greater the chance of distorting the signal being
isolated from the
noise.
Figure 7 illustrates one way to select rank k in order to remove as much noise
as
possible without distorting the signal of a given seismic data set 710.
Perform noise removal
730 separately for each of a suite of potential rank k values {e.g. 1 - 5).
For each result
calculate the difference between it and the input data, then plot this
difference making it easier
to quickly visualize how much of the signal has been removed. After comparing
the original with
the output difference for each of the k values applied, choose the smallest
rank whose
difference plot shows insignificant indications of signal (i.e. that looks
random with little
coherence). A person of skill in the art of seismic processing will readily
recognize when a
difference plot contains too much signal.
According to an alternate embodiment of the method of the present invention
non-
integer values of k may be applied to fine tune i:he signal to noise ratio of
the result. For
example, a K value of 2.7 may be implemented by zeroing out all but the three
largest singular
values, and multiplying the third largest singular value by .7 before the
matrix is recomposed. In
this circumstance k no longer represents rank, but rather a degree of noise
removal that is
intermediate that of rank 2 and rank 3.
The method of the present invention works equally well on both flat and
structured data
because this method does nothing to a noiseless seismic grid containing no
more than k dips,
which (unlike eigenimage filtering in the time domain) is because said method
operates in the
frequency domain of each trace.
According to a preferred embodiment of the method of the present invention not
all of
the frequency slices need to be rank reduced. Typically seismic traces are
sampled in time at a
rate such that the signal frequencies are a fraction of the Nyquist frequency.
For example, it is
common for seismic data to have significant signal only between frequencies 10
and 80 Hz (the
appropriate signal band is well known to persons of skill in the art of
seismic processing), yet the
Nyquist frequency is often 125 or 250 Hz - consequently only matrices 540
based on frequency
slices between 10 and 80 Hz need to be rank reduced. The remainder can be
ignored, left
unchanged, or zeroed each resulting in a considerable savings in computation.
CA 02402942 2002-09-12
9
Although the above described classical SVD works well for rank reduction, it
is
computationally expensive and therefore not recommended for use in the method
of the present
invention. Reasonable approximations to full decomposition can be computed
using Lanczos
bidiagonalization (O'Leary and Simmons, 1981; Simon and Zha, 2000), which can
require as
little as one-tenth the computation of the SVD method even though the quality
of results is
indistinguishable from the SVD results. Advantageously, when removing noise
from large data
sets, the method of the present invention can be executed much faster than the
closest known
competing method, being f-xy prediction filtering.
Advantageously, according to an alternate embodiment, the method of the
present
invention can handle erratic noise quite well by identifying the rank k
matrices that are near the
subject input matrix in an L1-norm sense using a robust SVD (Hawkins, Liu, and
Young,2001 ).
As illustrated in the "surface stacking diagram" of Figure 8, the method of
the present
invention can also be applied to unstacked 2-D seismic data sets when
unstacked traces 810
are laid out on a two-dimensional grid on which the trace shot (ordered by
increasing
receiver/station position) forms one axis 820 and the trace receiver forms the
other axis 830,
such that the data has the appearance of a stacked 3-D section permitting
noise removal to be
performed as set out above.
As illustrated in Figure 9, the method of the present invention can also be
applied to an
unstacked 3-D seismic data set. In a typical 3-D acquisition, shots 910 are
positioned spatially
along a multitude of "shot lines", and receivers 920 are positioned spatially
along a multitude of
"receiver lines". To perform noise removal according to an alternate
application of the method of
the present invention, extract all traces having been acquired on a single
shot line 930 and a
receiver line 940. These traces are then laid out on a spatial grid where
shots from the shot line
form one axis and receivers from the receiver line form the other axis -
giving the data the
appearance of a stacked 3-D section on which noise removal may be performed as
set out
above. The foregoing process is repeated for all remaining combinations of
shot lines and
receiver lines.
The method of the present invention works well for 2-D and 3-D unstacked data
sets
because: the method is independent of x- and y-consistent statics (i.e. The
Statics Property
according to Trickett, 2001 ); the method is exact for a noiseless seismic
grid that has no more
than k dips, and has then had x- and y-consistent filters applied (i.e. the
Filtering Property); and
if the method is exact for a seismic grid then the method is also exact the
same seismic grid
which has had rows or columns of traces duplicated or removed (i.e. the
Shooting Property).
Advantageously, as a result of the Statics and Filtering properties, and the
fact that the matrix
rows and columns are selected to represent common shots and receivers, the
random noise
removal can preserve surface-consistent effects, allowing the method to be
applied at a very
early stage of processing. To extract rectangular grids from unstacked 2-D and
3-D data sets for
noise removal, the x axis represents shots and y represents receivers because
then surface-
consistent (that is, shot and receiver) effects are left undistorted by the
method as a result of a
synergy between the method°s ability to absorb, or leave undistorted, x-
and y-consistent
effects, and the manner of extracting rectangular grids of traces from
prestack data sets.
Advantageously, the method of the present invention works well along a
straight spatial
boundary, since from the method's point of view there is no boundary, which
makes the method
well-suited for removing noise from common-offset or common-angle stacks, in
which many of
the traces are at or near a boundary. For common-offset or common-angle stacks
from a 2-D
CA 02402942 2002-09-12
acquisition, the traces are naturally laid out in a 2-D spatial grid, making
it possible to perform
noise removal as if it were a stacked 3-D section. Advantageously, the method
is independent
of the row and column ordering (as a result of the Ordering Property, pursuant
to Trickett, 2001 ).
According to an alternate embodiment of the method of the present invention
noise
reduction can be designed on one set of data, but applied on another. The
design data can be
taken from different time windows of the same traces as the application data,
or from a different
set of traces. This is made possible where matrix A holds the DFT values for a
given frequency
of the design data, and matrix C holds the DFT values for a given frequency of
the application
data - it is possible to calculate matrix B by projecting matrix C onto the
rank k subspace of
matrix A corresponding to its first k singular values.
According to an alternate embodiment of the method of the present invention,
by
applying different noise filters to the design data, it is possible to remove
coherent noise from
seismic data, as well as random noise, which permits tailoring the signal
subspace to avoid, and
thus remove, coherent energy,
The resulting signals may be transmitted to a remote location.
Although the disclosure describes and illustrates various embodiments of the
invention,
it is to be understood that the invention is not limited to these particular
embodiments. Many
variations and modifications will now occur to those skilled in the art of
processing seismic data.
For full definition of the scope of the invention, reference is to be made to
this disclosure
together with the appended papers authored by the Inventor, which papers form
a part of this
disclosure.
