CA2402942A1 - Method of using matrix rank reduction to remove random noise from seismic data processed in the f-xy domain - Google Patents

Description

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 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 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 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, 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.

(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 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 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.

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

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.

Claims