Method of processing seismic data

A method of processing seismic data wherein seismic data signals having a first part which shows a slow variation in offset and a second part which shows a slow variation in midpoint, is modelled using a model which allows the two parts to be separated. The individual parts may be reconstructed from the model. One of the parts corresponds to offset dependent noise and can be effectively removed from the seismic data.

BACKGROUND OF THE INVENTION 
The present invention relates to a method of processing seismic data. Such 
a method may be used for marine seismic data or land seismic data and for 
two dimensional or three dimensional data. 
In seismic exploration, seismic data are gathered by actuating one or more 
sources of seismic energy and recording at one or more locations reflected 
seismic energy so as to provide information about the reflective 
boundaries below the surface of the earth. For instance, in the case of a 
typical marine seismic exploration, a seismic source comprising one or 
more air guns or the like and a seismic streamer comprising hydrophones 
spaced regularly along a cable are towed behind an exploration vessel. The 
source is periodically actuated or fired and the reflected pressure 
variations detected by the hydrophones are recorded as individual traces. 
The data are therefore in the form of a large number of traces which are 
stored and subsequently processed using various techniques. 
The traces are normally re-ordered or arranged into common mid-point 
gathers. The midpoint is the middle of a straight line extending from the 
source to the receiving transducer for the trace, so that all traces 
having the same mid point are grouped together and are sorted as a 
function of offset i.e. the distance between the source and the receiver. 
The travel times of the incident and reflected waves vary for different 
offsets, and this is corrected by means of normal move out (NMO) 
correction. Optionally, dip move out (DMO) correction is applied so as to 
compensate for non-horizontal reflective boundaries beneath the surface of 
the earth. The corrected traces are then "stacked" i.e. added together for 
each common mid-point gather, resulting in one stacked trace per common 
mid-point position. 
For a three dimensional exploration giving information about a volume of 
the earth below the surface, the reflected seismic wave field may be 
considered as a five dimensional space having two spatial coordinates for 
the mid-point, one spatial coordinate for the offset, one spatial 
coordinate for the offset direction or azimuth, and a time coordinate. 
Instead of offset and azimuth coordinates, the offset may be defined in 
terms of spatial i.e. cartesian coordinates. For a two-dimensional 
exploration of a vertical slice through the earth, the mid-point and 
offset may each be defined by a single spatial coordinate so that three 
dimensional space is sufficient. The seismic wavefield is normally sampled 
at regular discrete time intervals and irregular discrete mid-points and 
offsets. 
The reflected seismic wave field after NMO correction may comprise three 
parts. The first part depends on the mid-point position and varies only 
slowly with offset. A second part depends on the offset and varies only 
slowly with mid-point position. The third part comprises the rest of the 
seismic wavefield. 
All three parts of the seismic wavefield contain useful information. The 
first part provides useful seismic information about the structure of the 
earth. The second part comprises offset-dependent or shot-generated noise 
whereas the third part contains all other types of noise. The second part 
includes the major types of noise in the seismic wavefield, such as noise 
from waves which propagate horizontally along the surface e.g. "ground 
roll" for land data. 
BRIEF SUMMARY OF THE INVENTION 
It is an object of the invention to provide a method of processing a 
plurality of input seismic data samples having a plurality of mid-points 
and a plurality of offsets. This method includes selecting a sample 
parameter of each of the input seismic data samples, defining at least one 
modeling function having at least one parameter, a first part which is 
substantially independent of the offsets of the input seismic data samples 
and a second part which is substantially independent of the mid-points of 
the input seismic data samples, and selecting the at least one parameter 
of the at least one modeling function so that the at least one modeling 
function represents a best fit to the selected sample parameters.

FIG. 1 of the accompanying drawings illustrates the different parts of the 
recorded seismic data for a two dimensional exploration. This drawing 
illustrates the mid-point wave number against the offset wave number 
corresponding to one time slice or frequency component in the two 
dimensional measured wavefield. For a three dimensional exploration, both 
the midpoint wave number axis and the offset wave number axis would have 
to be extended to two dimensions. The first part 1 of the wavefield is 
located around the mid point wave number axis with small offset wave 
numbers corresponding to slow variations in the offset direction. An 
offset wave number equal to zero corresponds to no variation against 
offset. The part 2 of the wave field representing offset-dependent noise 
is located around the offset-wave number axis with only small mid-point 
wave numbers corresponding to slow variations in the mid-point direction. 
FIG. 2 of the accompanying drawings illustrates a special case in which the 
signal part 1 is independent of offset and the noise part 2 is independent 
of midpoint. For this special case, the rectangles shown in FIG. 1 reduce 
to line segments on the respective axis. 
FIG. 3 of the accompanying drawings illustrates a case where DMO correction 
has not been applied. For larger mid-point wavenumbers, no perfect 
alignment in the offset direction occurs. The signal is thus contained in 
the "bow tie" shaped part 1 of FIG. 3. 
The stacking of common midpoint traces does not affect the signal part of 
the data because this part does not vary with offset. In the case of a 
common midpoint gather containing a continuous range of offsets, the 
offset dependent noise is cancelled by the stacking operation so that it 
is substantially suppressed. Thus, the signal-to-noise ratio of stacked 
data is improved over unstacked data. However, if a common midpoint gather 
does not contain a continuous range of offsets, the offset-dependent noise 
is not fully cancelled so that a substantial amount of this noise remains 
in the stacked traces. The phenomenon is caused by a so-called "leaking 
stack operator". 
An example of "sparsely sampled data sets" with a non-continuous offset 
range for each common mid-point gather occurs in two dimensional 
exploration where the source interval is different from the receiver 
interval. If the source spacing or interval is different from a multiple 
of the receiver spacing or interval, the spacing between the mid-points is 
smaller than half the receiver interval. Further, for three dimensional 
exploration, all three dimensional data for land as well as for marine 
data are sparsely sampled in the offset direction. 
According to the invention, there is provided a method of processing a 
plurality of input seismic data samples having a plurality of mid-points 
and a plurality of offsets, by selecting a sample parameter of each of the 
input seismic data samples; defining at least one modelling function 
having at least one model parameter and comprising a first part which is 
substantially independent of the offsets of the input seismic data samples 
and a second part which is substantially independent of the mid-points of 
the input seismic data samples; and selecting the at least one parameter 
of the at least one modelling function such that the at least one 
modelling function represents a best fit to the selected sample of the 
input seismic data samples. 
The at least one modelling function may comprise at least one first 
modelling function which is substantially independent of the offsets of 
the input seismic data samples and at least one second modelling function 
which is substantially independent of the mid-points of the input seismic 
data samples. 
In a first embodiment the selected samples may comprise the value of each 
input seismic trace at a predetermined time. In a second embodiment, the 
selected sample may comprise temporal Fourier coefficients of the input 
seismic data, thereby selecting input seismic data at a predetermined 
frequency. 
The modelling functions may comprise sinc functions. As an alternative, the 
modelling functions may comprise sine and/or cosine functions. In one 
embodiment, sinc functions may be used as the first modelling functions 
for common offset traces whereas the second modelling functions for the 
common mid-point traces may comprise sine and cosine functions. Further, 
irregular mid-points and offsets may be replaced by spatially regularly 
sampled data by employing the techniques disclosed in British Patent 
Application No. 9111145.0 filed on May 23, 1991. 
The step of selecting the first and second parameters amounts to a 
"parametric inversion scheme" and any suitable scheme may be employed, 
such as the well-known least squares inversion where the first and second 
parameters are chosen such that the sum of the squares of the differences 
between the modelled parameters and the selected sample parameters are 
minimized. 
Such a method may be applied to two dimensional data and two three 
dimensional data, and may be preceded by NMO correction and DMO 
correction. 
The method may further comprise constructing output data samples from the 
at least one first modelling function. The output data samples may be used 
in place of the input seismic data samples and are substantially free from 
offset-dependent noise. 
The method may further comprise constructing offset dependent noise samples 
from the at least one second modelling function. The offset-dependent 
noise samples may be subtracted from the input seismic data samples so as 
to remove the effects of offset-dependent noise. 
The output data samples and the offset-dependent noise samples may be 
subtracted from the input seismic data samples so as to produce non-offset 
dependent noise samples, which may contain information of interest. 
The method may further comprise stacking the output data samples having 
common mid-points. Other processing techniques may then be applied, such 
as pre-stack or post-stack migration. 
It is thus possible to provide a method which is capable of separating the 
different parts of the seismic response from sparsely sampled data. Such a 
method uses all traces in an area to form a continuous range of offsets 
and uses the actual mid-point positions of these traces to separate the 
offset-dependent part of the data from the part which is independent of 
offset. By using the continuous range of offsets in this area, the 
offset-dependent noise can be effectively eliminated. 
The present invention will be further described, by way of example, with 
reference to FIGS. 4 to 6 of the accompanying drawings, in which: 
FIG. 4 illustrates a stack, based on synthetic input data for a sparsely 
sampled seismic survey with offset-dependent noise and a signal comprising 
eleven reflections, illustrating the effect of a leaking stack operator; 
FIG. 5 illustrates a signal determined from the synthetic input data by a 
method constituting an embodiment of the invention; and 
FIG. 6 illustrates offset-dependent noise determined from the synthetic 
input data by a method constituting an embodiment of the invention. 
Seismic data acquired during exploration are arranged into common mid-point 
gathers, each of which contains a plurality of recorded traces with 
discontinuous irregularly spaced offsets. The traces in each common 
mid-point gather are then subjected to NMO correction and optionally to 
DMO correction. 
Each of the corrected samples representing the seismic wavefield has a part 
which varies only slowly with offset and represents the signal part. The 
offset-dependent part of each sample which is substantially independent of 
mid-point represents the offset-dependent noise. In order to separate the 
signal and noise parts, it is assumed that the signal part is 
substantially independent of offset and depends only on the geology of the 
part of the earth being explored. Thus, the signal part is dependent on 
the common mid-point position with only smooth variations in the offset 
direction. Similarly, the offset-dependent part is substantially 
independent of common mid-point position with only smooth variations of 
noise with respect to common mid-point position. 
It is thus possible to construct a "forward model" which describes data 
composed of signal and offset-dependent noise with a plurality of 
modelling functions containing selectable parameters. By appropriately 
selecting the parameters, the modelling functions can be used to calculate 
the data for any mid-point position and offset in the area and offset 
range of interest. The model makes estimates of the input traces which 
have been recorded and corrected at the actual mid-point positions and 
offsets of these traces, which may be irregularly sampled. The estimated 
traces are compared with the input traces and the parameters of the 
modelling functions are adjusted so as to obtain the "best fit" or 
approximation to the recorded and corrected data. 
Various modelling functions may be used in order to construct the forward 
model. For instance, the modelling functions may comprise sinc-functions 
at different regular positions, in which case, the parameters are the 
amplitudes of the sinc-functions at regular mid-point positions or 
offsets. Alternatively, the modelling functions may be sine or cosine 
functions with different frequencies. The parameters then comprise the 
Fourier coefficients in the mid-point or offset domain. The modelling 
functions may comprise a combination of two different types of functions, 
for instance with one type, such as the sinc functions, describing the 
signal part and the other type, such as the sine and cosine functions, 
describing the noise part. 
The arguments of the modelling functions depend on the possibly irregular, 
mid-point positions and offsets of the measured seismic traces. 
Thus, parameters of the forward model can be estimated from the input 
traces using the actual mid-point positions and offset positions, which 
may be irregularly sampled. The parameters are chosen such that the 
estimates produced by the model provide the best fit to the actual input 
traces. The instantaneous amplitudes or values of the input traces at 
predetermined times corresponding to a time slice through the input 
traces, or the temporal frequency components, are estimated and compared 
with the actual values. The process is repeated for each time slice or 
each frequency component so as to build up a complete estimate of the 
wavefieid. These techniques may also be applied to a number of time slices 
or frequencies simultaneously. 
The parameters of the forward model are estimated by any suitable 
parametric inversion scheme. In such schemes, the mis-match between the 
actual data and the data modelled with the currently chosen parameters is 
minimised. For instance, in the well-known least squares inversion 
algorithm, the sum of the squared difference between input data and 
estimated data is minimised. The modelling functions which are chosen to 
fit the input data have a wave number spectrum comprising the parts 1 and 
2 shown in any of FIGS. 1 to 3. 
The forward model can be written as a matrix multiplication: 
EQU y(x,h)=A.sub.1 g.sub.s +A.sub.2 g.sub.n 
where x is midpoint position, h is offset, y(x,h) is a vector containing 
the seismic data for all input traces, g.sub.s is a vector containing the 
parameters of a first (signal) modelling function, g.sub.n is a vector 
containing the parameters of a second (noise) modelling function, the 
matrix A.sub.1 contains the signal modelling function whose arguments 
depend on midpoint positions of the measured seismic data, and the matrix 
A.sub.2 contains the noise modelling functions whose arguments depend on 
the offsets of the measured seismic data. 
The two matrix multiplications can be combined into one matrix 
multiplication: 
##EQU1## 
where g now contains the parameters of both the first and second modelling 
functions. 
Using the least squares inversion, the parameters can be determined by: 
##EQU2## 
thus separating the estimated signal parameters g'.sub.s from the 
estimated noise parameters g'.sub.n. Output traces representing the signal 
and the off-set dependent noise can then be constructed from the 
parameters g'.sub.s and g'.sub.n, respectively. 
It is thus possible to extract from input seismic data samples the signal 
part (FIG. 5) of the seismic wavefield and the offset-dependent noise 
(FIG. 6). In particular, for each combination of mid-point position and 
offset, the signal and offset dependent noise contributions can be 
calculated separately. This information may then be used in various 
different ways. For instance, a signal can be estimated which varies with 
mid-point and also varies slowly with offset. Such data can then be 
further processed as prestack data prior to stacking. 
The signal as a function of mid-point position can be generated and used 
for further processing. Such data are comparable to the standard stacked 
data, where the effect of the leaking stack operator has been eliminated. 
Further, such data can be generated for regular spatial sampling 
irrespective of whether the input data was irregularly sampled. 
The offset-dependent noise can be subtracted from the original input 
traces. This provides prestack data with a better signal-to-noise ratio, 
which may then be subjected to further prestack processing. Also, it is 
possible to examine the offset-dependent noise separately. 
In a further technique, the estimated signal and the estimated 
offset-dependent noise can be subtracted from the original input data. The 
result of this is to reveal all of the noise which is not modelled by the 
forward model. Analysis of the signal, offset-dependent noise, and other 
noise may give more insight into the information content of the acquired 
seismic data. 
These techniques may be applied both to two dimensional and to three 
dimensional seismic data and to both marine and land seismic data. The 
techniques may be applied immediately after NMO correction and, when 
performed, DMO correction, with the signal part in the form of output 
traces being used in subsequent processing in place of the corrected input 
traces. The techniques of standard migration and prestack migration may be 
applied to the output data, or the data may be stacked and then subjected 
to post stack migration.