Method of reverberation removal from seismic data and removal of dual sensor coupling errors

A method for eliminating the influence of multiple reflections in seismic data by determining an up going and down going vector wave-field from the vector wave-field, determining a product of the free surface reflection coefficient and the down going vector wavefield, and adding the product to the up going vector wave-field. Also provided is a method for eliminating the effects of receiver coupling from seismic data taken in a survey by describing a first cross-equalization filter as a function of the reverberation period, describing a second cross-equalization filter as a function of the seismic data, describing an inverse coupling filter as a function of the first cross-equalization filter and the second-equalization filter, and applying the coupling filter to the seismic data.

BACKGROUND OF THE INVENTION 
This invention relates to the field of seismic data acquisition and 
processing. In particular, this invention relates to a method for removing 
multiple reflections from seismic data. It also relates to removal of 
coupling errors in dual sensor data. 
The problems of multiple reflections in seismic data is well known, and 
many attempts to remove or lessen its influence on data have been made. 
See, for example, U.S. Pat. Nos. 4,979,150; 5,163,028; 5,235,554; and 
5,365,492 all of which are incorporated herein by reference. It will be 
noted that the above patents are focused on the problem of free surface 
multiples caused by reverberations in a water column; however, the free 
surface multiple reflections can and do occur from reverberations between 
reflecting interfaces in the earth, also. These four patents cite many 
others considered relevant for one reason or another, for example, U.S. 
Pat. Nos. 2,757,356; 3,290,645; 3,299,397; 3,943,484; 3,979,713; 
4,134,097; 4,146,871; 4,253,164; 4,296,481; 4,348,749; 4,380,059; 
4,437,175; 4,477,887; 4,486,865; 4,520,467; 4,581,724; 4,644,507; 
4,644,508; 4,658,387; 4,685,090; 4,733,379; 4,736,345; 4,752,916; 
4,821,241; 4,910,716; 4,956,822; 5,251,183; and 5,257,241; all of which 
are incorporated herein by reference. Further, the following articles may 
also be considered pertinent in evaluation of the present invention: Monk, 
D. J., Wavefield Separation of Twin Streamer Data, First Break, Vol. 8, 
No. 3, March, 1988, pgs. 96-104; Brink, M., Application of Vertical 
Receiver Arrays in 3D Seismic Exploration, Society of Exploration 
Geophysicists, 1988, pgs. 460-463; Wuenschel, P. C., Removal of the 
Detector-Ground Coupling Effect in the Vertical Seismic Profiling 
Environment, Geophysics, Vol. 53, No. 3, March, 1988, pgs. 359-364; Bell, 
D. W., et al., Two-Trace Directional Filter For Processing Offset Vertical 
Seismic Profiles, AAPG Bulletin, Vol. 72, No. 3, March 1988, pg. 375; 
Brink, M. et al., Marine Seismic Exploration Using Vertical Receiver 
Arrays: Acquisition in Bad Weather, 49th Meeting of European Assn. of 
Exploration Geophysicists, June 1987; Tan, T. H., Reciprocity Theorem 
Applied To the Geophone-Ground Coupling Problem, Geophysics, Vol. 52, No. 
12, December 1987, pgs. 1715-1717; Krohn, C. E., Geophone Ground Coupling, 
Geophysics, April, 1985, pgs. 56-60; Krohn, C. E., Geophone Ground 
Coupling, Geophysics Vol. 49, No. 6, June 1984, pgs. 722-731; Plane-wave 
Decomposition of Seismograms, Geophysics, Vol. 47, No. 10, October, 1982, 
pgs. 1375-1401; A. M. Ziolkowski, Source Array Scaling for Wavelet 
Deconvolution, Geophysical Prospecting, Vol. 28, No. 6, December 1980, 
pgs. 902-918; A. M. Ziolkowski, Wavelet Deconvolution Using a Source 
Scaling Law, Geophysical Prospecting, Vol. 28, No. 6, December 1980, pgs. 
872-901; G. M. Hoover, J. T. O'Brien, The influence of the planted 
geophone on seismic land data, Geophysics, Vol. 45, No. 8, August 1980, 
pgs. 1239-1253, J. White, Chapter 2--Plane Waves, Seismic Wave 
Radiation--Transmission and Attenuation, Seismic Waves, McGraw Hill 
Publish., 1965, pgs. 15-41, B. Widrow, J. Glover, Jr., J. McCool, J. 
Kaunitz, C. Williams, R. Hearan, J. Zeidler, E. Dong, Jr., R. Goodlin, 
Adaptive Noise Cancelling: Principles and Applications, Proceedings of the 
IEEE, Vol. 63, No. 12, December 1975, pgs. 1692-1716; H. Washburn, H. 
Wiley, The effect of the placement of a seismometer on its response 
characteristics, Presented at the Annual Meeting, Chicago, Apr. 11, 1940. 
In addition, the following U.K Patents were cited in U.S. Pat. No. 
4,979,150, which may be redundant to other cited U.S. patents: Ruehle, 
U.K. Patent No. 1316479, Nov. 23, 1970; Broding, U.K. Patent No. 2004648, 
Apr. 4, 1979; and Hutchins, U.K. Patent No. 2030400, Apr. 2, 1980. 
Other references found when searching in a related area include the 
following U.S. Pat. Nos., which are incorporated herein by reference: 
4,794,572; 4,803,666; 4,817,061; 4,888,743; 4,903,244; 4,912,979; 
4,933,913; 4,935,903; 5,027,332; 5,029,146; 5,136,554; 3,350,683; 
3,689,874; 4,234,938; 4,935,903; 4,937,793; 4,992,993; 5,163,028; 
5,235,554; 5,365,492; and 5,396,472. 
Generally, these methods require calibration shooting or estimates of water 
bottom reflectivity. Calibration shooting is expensive and can introduce 
its own errors, while estimates of water bottom reflectivity are 
inherently inaccurate. Current statistical methods are flawed by noise. 
Further, imperfect coupling between a geophone and the earth and other 
response differences between co-located hydrophone and geophone pairs 
(a.k.a. "dual sensors" to those of ordinary skill) exist which are 
different from pair to pair. Application of the same correction scheme to 
each dual sensor pair is not an optimum solution. 
Accordingly, there is a need for a method for eliminating free surface 
multiples from seismic data where there exists a free surface reflection 
coefficient. Further, there is a need for compensation for coupling 
differences between each sensor in a dual sensor pair. 
SUMMARY OF THE INVENTION 
It is an object of the present invention to address the above needs, in one 
embodiment, by a method comprising: determining an up going and a down 
going vector wave-field from the seismic data, determining a product of 
the free surface reflection coefficient and the down going vector 
wave-field, and adding the product to the up going vector wave-field. It 
has been found that such a method avoids the problems inherent in the 
previous attempts to eliminate free surface multiples, which were caused 
by combining particle velocity and pressure data. 
According to another aspect of the present invention, a method is provided 
for eliminating the effects of receiver coupling from seismic data taken 
in a survey, wherein there exists a reverberation response period. The 
method comprises: describing a first cross-equalization filter as a 
function of the reverberation period; describing a second 
cross-equalization filter as a function of the seismic data; deriving an 
inverse coupling filter as a function of the first cross-equalization 
filter and the second-equalization filter; and applying the coupling 
filter to the seismic data.

DETAILED DESCRIPTION 
Referring now to a specific example embodiment of the present invention, it 
will be recognized that receiver ghosts are recorded on seismic data when 
a receiver, located in the water column, senses reflection energy which is 
reverberating in the water column. The present invention eliminates the 
free surface multiples by separating the up going and down going vector 
wave-fields (U(Z) and D(Z) respectively), then adding the product of the 
down going wave-field U(Z). 
A model for the P-wave energy reverberating in the water column was 
outlined by Milo Backus in 1958. See, Water Reverberations--Their Nature 
and Elimination, Geophysics, 1958) incorporated herein by reference. As 
shown in FIG. 1, P-wave reflection energy 10 arrives in the water column 
12 from depth, then bounces between the water surface 14 and the water 
bottom 16. The relative polarity and amplitude of the P-wave 10 for any 
given point in time is determined by the product of the reflection 
coefficients for each successive bounce between the water surface and 
water-bottom. For detectors located on the water bottom, the pressure 
response P(Z) and velocity response V(Z) are described by equations (1) 
and (2). 
##EQU1## 
where P=pressure 
V=particle velocity 
Z=e.sup..iota..omega..tau. 
.alpha.=impedance 
.theta.=angle of incidence 
r=reflection coefficient of the water bottom 
##EQU2## 
d=vertical water depth .nu.=water velocity 
Calculating the closed form of equations (1) and (2) gives equations (3) 
and (4). 
##EQU3## 
The up going vector wave-field, U(Z), is determined by adding equations 
(3) and (4). The down going vector wave-field, U(Z), is determined by 
subtracting (3) from (4) (See, Lowenthal and Gardner, 1985), and 
Lowenthal's U.S. Pat. No. 4,752,916, both of which are incorporated herein 
by reference). Equations (5) and (6) represent the up going and down going 
components on an infinite series of water reverberations. 
##EQU4## 
This infinite series of water reverberations can be eliminated by taking 
the product of the down going wave-field D(Z) and adding it to the up 
going wave-field U(Z). 
##EQU5## 
where 
EQU -1.ltoreq.r.ltoreq.+1 
This technique is demonstrated on a zero-offset model shown in FIGS. 2-7. 
FIG. 2 shows a band limited spike at 40 milliseconds. FIGS. 3 and 4 show 
the pressure and velocity response for detectors on the water bottom at 30 
meters water depth. FIGS. 5 and 6 show the separated up going and down 
going vector wave-fields. FIG. 7 shows the de-reverberated wave-field by 
adding the product of an estimated water bottom reflection coefficient and 
the down going vector wave-field to the up going vector wave-field. 
In summary, reverberation-free primary P-wave data is achieved in various 
embodiments of the invention by recording pressure and particle velocity 
data, separating the up going and down going vector wave-fields, and 
adding the product of the water bottom reflection coefficient and the down 
going vector wave-field to the up going vector wave-field. 
Many ways of determining the up going and down going vector wave-fields are 
acceptable. For example, one method includes collocating seismic data 
receivers at the free surface, either physically, or mathematically. Also, 
the free surface is defined to be a reflecting interface such as, for 
example, an air water interface or the water bottom in the case of water 
column reverberations in marine seismic data. The free interface also may 
be defined between geologic layers to reduce reverberations that occur in 
the earth's structure. 
In another example, the determining an up going and a down going vector 
wave-field further comprises the step of describing the seismic data as a 
function of the free surface reflection coefficient and a reverberation 
period in an expanded form. 
Further, in some embodiments, the data is collected during the survey with 
co-located pressure and particle velocity response receivers, which are 
commonly known in the art. In other embodiments, the data is collected 
with vertically-spaced pressure receivers. 
Alternatively, in another example, the determining an up going and a down 
going vector wave-field further comprises the step of describing the 
seismic data as a function of the free surface reflection coefficient and 
a reverberation period in a closed form. 
In one implementation of this aspect of the invention, the seismic data 
comprises a first set of seismic data in the closed form and a second set 
of seismic data in the closed form and the first set of seismic data is 
added to the second set of seismic data, wherein the up going vector 
wave-field is defined. Alternatively, the first set of seismic data is 
subtracted from the second set, wherein the down going vector wave-field 
is defined. 
As mentioned above, receiver coupling is also a factor in accuracy of 
seismic surveys in Dual Sensor bottom reference receiver acquisition 
(DSSR), where a pressure detector and a particle velocity detector are 
co-located on the water bottom. A seismic source is fired and the 
returning reflection energy is recorded by the two detectors. The two 
recorded data sets are summed and subtracted to separate the up going and 
down going vector wave-fields. A fundamental issue with this technique is 
the coupling of the particle velocity detectors to the water bottom. 
According to one aspect of the present invention, therefore, a process is 
provided by which the pressure detector is used as a guide function to 
estimate the coupling degradation of the recorded particle velocity data. 
An inverse filter is derived and applied to the particle velocity data 
prior to vector wave-field separation. 
For pressure and particle velocity detectors co-located on the water 
bottom, the reverberation response is given by equations (1a) and (2a). 
##EQU6## 
where P=pressure 
V=particle velocity 
Z=e.sup..iota..omega..tau. 
.alpha.=impedance 
.theta.=angle of incidence 
r=reflection coefficient of the water bottom 
.tau.=two-way travel time through the water column 
For the purposes of this discussion, we will be looking at the zero-offset 
case for which cos .theta.=1. We will assume a correction factor has been 
applied to the particle velocity data to remove .alpha.. 
A filter, X(.omega.), which will convert the pressure data into particle 
velocity data, can be described in the frequency domain as 
EQU P(.omega.)X(.omega.)=V(.omega.) (3a) 
Solving for X(.omega.). 
##EQU7## 
Substituting the frequency domain expressions of equations (1a) and (2a) 
into equation (4a) yields equation (5a). 
##EQU8## 
Solving for the amplitude and phase components of equation (5a) gives 
equation (6a). 
##EQU9## 
Thus, for co-located pressure and particle velocity detectors on the water 
bottom, the filter X(.omega.) which converts pressure data into particle 
velocity data has an amplitude component which is solely dependent on the 
period of the water reverberations .tau., and a constant phase component 
of 90 degrees. Imperfect coupling of the particle velocity detector to the 
water bottom can be expressed as a filter, c(.omega.), applied to the 
particle velocity data which distorts the amplitude and phase of the data. 
Thus a cross-equalization filter, X.sub.c (.omega.) calculated from the 
recorded data will have this filter applied to the particle velocity 
field. 
##EQU10## 
The ideal cross-equalization filter X(.omega.), without c(.omega.), can be 
calculated with a priori knowledge of .tau. using equation (6a). Dividing 
X(.omega.), by X.sub.c (.omega.), results in the inverse coupling filter. 
##EQU11## 
This filter can be applied to recorded particle velocity data in order to 
remove the coupling effects. 
In summary, an ideal cross-equalization filter, for pressure and particle 
velocity detectors co-located on the water bottom, is calculated from 
knowledge of the period of the water reverberations. This ideal 
cross-equalization filter is compared to the cross-equalization filter 
calculated from the recorded data. The result of this comparison is an 
inverse filter which, when applied to the recorded particle velocity data, 
removes the effects of receiver coupling. 
Other embodiments of the present invention will occur to those of skill in 
the art which do not depart from the spirit of the invention.