Method for improving the coupling response of a water-bottom seismic sensor

A receiver consistent deconvolution operator models the damped oscillatory wavetrain that is related to geophone coupling to the water bottom. The operator is a best-fitting function that endeavors to describe the difference in coupling response between a well-coupled in-line geophone relative to an imperfectly-coupled cross-line geophone. The operator is applied to the cross-line signals to compensate the signals for the distortion due to imperfect cross-line ground coupling.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
This invention is concerned with improving the coupling response of a 
multi-axis seismic sensor or geophone implanted on a water bottom with 
particular attention to the sensor whose axis of sensitivity is spatially 
polarized along the cross-line axis. 
2. Discussion of Relevant Art 
Although the basic principles of seismic exploration are well known, a 
brief tutorial exegesis of the geophysical problems to be addressed by 
this invention now will be presented. 
Please refer to FIG. 1 where a line of seismic transducers 10.sub.0, 
10.sub.1, 10.sub.2, . . . , 10.sub.s (s=3, 4, . . . , n, where n is a 
large integer) are shown laid on the bottom 12 of a body of water, 14 
(which may be for example, the open ocean, a lake, a bay, a river, a 
reservoir) spaced-apart by a desired spacing such as 25 meters. For 
purposes of this disclosure, the transducers are multiaxial 
motion-sensitive devices. In this disclosure, the terms "sensor", 
"receiver", "geophone" are synonymous. The terms refer to a 
mechanical-motion transducer which measures particle velocity. It to be 
distinguished from a hydrophone which is a pressure transducer. 
The sensors are mechanically and electrically coupled to an ocean-bottom 
cable 16, one or both ends of which may be marked at the water surface 11 
by a buoy such as 18. In practice, the cable 16 may be hundreds or 
thousands of feet long to which are attached many hundreds of sensors. For 
3-D areal surveys, a number of cables may be laid out parallel to each 
other in a wide swath. 
Usually, the cables and sensors are laid out over the area to be surveyed 
by a cable-tender boat. At some later time, a service ship such as 20 
visits designated stations and retrieves one or more cables such as 16 
from the water bottom, the ends having been flagged by buoys as shown. 
Cable 16 includes a plurality of internally-mounted communication channels 
(not shown), that may be electrical, optical, or in some cases ethereal, 
for transmitting the sensor output signals to digital data-recording and 
data-processing channels of any well known type (not shown) in ship 20. 
Ship 20 is equipped with a precision navigation means such as a GPS 
receiver and may include a radar beacon 22 for ranging on a radar 
reflector 24 mounted on tail buoy 18 at the other end of cable 16. 
An acoustic sound source is fired at selected shot locations which may be 
spaced apart by an integral multiple of the sensor spacings. Source 26 
radiates wavefields such as generally shown by 28 and 30 to insonify 
subsurface earth layers such as 32, whence the wavefield is reflected back 
towards the surface as reflected wavefield 34. The sensors 10.sub.s 
intercept the mechanical earth motions, convert those motions to 
electrical signals and send those signals through the communication 
channels to the recording equipment in ship 20. 
A wavefield may propagate along a direct travel path such as 36 or along 
reflected-ray travel paths such as 38, 38' and 38" to the respective 
sensors 10.sub.s. The recorded data are presented in the form of 
time-scale traces, one trace per sensor/shot. A collection of time-scale 
traces resulting from a single source activation (a shot) that insonifies 
a plurality of receivers, such as in FIG. 1, constitutes a common source 
gather. On the other hand, with reference to FIG. 5, a collection of 
time-scale traces as recorded by a single sensor 10.sub.s after 
insonification by a plurality of spaced-apart shots 26, 26', 26" 
constitutes a common receiver gather. The space between a shot location 
and the surface expression, 27, of sensor 10.sub.s is the offset. 
Typically in 3-D operations ship 20 occupies a central location, 
interconnected with a plurality of receivers, while a second shooting ship 
(not shown) actually visits the respective designated survey stations to 
generate common receiver gathers. The practice is necessarily required in 
3-D because the survey stations are scattered over a two-dimensional area 
rather than being restricted to a single line of profile. 
FIG. 2 is a close-up, X-ray-like side view of a multi-axis motion sensor 
10.sub.s. The sensitive axes may be vertical, unit 40; in-line, unit 42; 
cross-line, unit 44. Usually, the two horizontally-polarized sensors 
preferably respond to shear waves and the vertical sensor responds to 
compressional waves. In some cases a two-axis instrument may be used for 
detecting shear waves only, with the sensor units directionally polarized 
along orthogonal x and y axes. 
The multi-axis units are customarily packaged in a single case and 
internally gimbal-mounted so as to become automatically aligned along 
their mutually orthogonal axes after deposition on the sea floor. For good 
and sufficient reasons, the case containing the sensor components is 
usually cylindrical. Cable 16 is relatively heavy. Secured to the fore and 
aft ends of the sensor case, the cable 16 firmly holds the multi-axis 
motion sensor to the sea floor 12. The in-line unit 42 is well coupled to 
sea floor 12 because it is oriented in the direction of the cable 16. In 
this direction, the area of contact with the sea floor is relatively 
large. Not so, the cross-line unit. 
FIG. 3 is an X-ray-like cross section of multi-axial sensor 10.sub.s taken 
along line 3-3', looking back towards ship 20. Because of its cylindrical 
shape, case 10.sub.s not only rolls from side to side as shown by curved 
arrows 46, but water currents and other disturbances can cause the sensor 
to shift laterally in the cross-line direction as shown by arrows 48, 48'. 
Those disturbances do not affect the in-line units because of their 
respective polarizations but they do introduce severe noise to the 
cross-axis signals. 
FIG. 4 is multi-axis sensor 10.sub.s as viewed from above along line 4-4' 
of FIG. 2. This Figure will be referenced again later. 
A geophone as used on ocean-bottom cables is a spring-mass oscillatory 
system. Assuming perfect coupling, the transfer function of a geophone can 
be described in terms of damping, .eta., resonant or natural frequency, 
.omega., and phase angle, .phi., relative to an input step function. 
Customarily geophones are damped at about 0.7 of critical at a resonant 
frequency of about 10-20 Hz. Assuming use of a velocity phone, below the 
natural frequency, the attenuation rate is 12 dB per octave; well above 
the natural frequency, the response is substantially constant within the 
useful seismic frequency band. The phase response may be non-linear below 
the resonant frequency and lags about 90.degree. behind the input 
transient above that value. Other signal distortions may be superimposed 
on the sensor output signals due to the respective transfer functions 
characteristic of the data transmission channels and the data processing 
equipment. 
Instrumental response parameters can, of course be predicted on the basis 
of design criteria. But an imperfect earth-coupling response cannot be 
predicted. Multi-axis seismic sensors are essential for use in shear-wave 
surveys where, for example, in-line and cross-line shear waves are 
resolved to measure the azimuth of substantially-vertical formation 
fracturing. It is evident that if the cross-line sensor response is 
distorted relative to the in-line sensor response, the resulting azimuth 
determination will be flawed. Resource-exploitation operations premised on 
flawed data is doomed to economic catastrophe. 
A method for correcting poor coupling of a logging sonde in a borehole was 
described in a paper by J. E. Gaiser et al., entitled Vertical Seismic 
Profile Sonde Coupling, published in Geophysics n. 53, pp 206-214, 1988. 
However that method is not directly applicable to 3-D seismic exploration. 
There is a long-felt need for a method for measuring and suppressing 
signal distortion attributable to poor water-bottom coupling of one of the 
components of a cable-mounted, multiaxial sensor. 
SUMMARY OF THE INVENTION 
This is a method for removing objectionable ground-coupling response 
characteristics from seismic signals due to an imperfect ground coupling 
of a seismic receiver that is polarized in the cross line direction 
relative to a well-coupled, co-located seismic receiver polarized in the 
in-line direction. Along a preselected source-receiver trajectory vector, 
a plurality of in-line seismic-signal wavetrains emanating from an in-line 
receiver is assembled into a first common-receiver trace gather. Similarly 
a plurality of cross-line seismic-signal wavetrains emanating from a 
seismic receiver that is co-located with the in-line receiver are gathered 
in a second common-receiver trace gather. Each of the seismic-signal 
wavetrains resident in the respective first and second common-receiver 
trace gathers is auto-correlated in the time domain to provide a plurality 
of in-line and cross-line auto correlations. The respective in-line 
auto-correlations are normalized to unity and the respective cross-line 
auto-correlations are normalized to the corresponding in-line 
auto-correlations. The normalized cross-line auto-correlations are scaled 
to compensate for the difference between the cross-line polarization 
direction and the pre-selected source-receiver trajectory vector. The 
normalized in-line auto-correlations and the normalized, scaled cross-line 
auto-correlations are averaged and the averages are transformed to the 
frequency domain to define in-line and cross-line amplitude spectra. The 
cross-line auto-correlation is deconvolved by the in-line auto-correlation 
to define a coupling deconvolution operator. The coupling deconvolution 
operator is applied to the cross-line seismic-signal wavetrains resident 
in the cross-line common-receiver trace gather to remove the imperfect 
ground-coupling response characteristics from the cross-line receiver 
signals.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
A deconvolution operator is desired that forms a receiver-consistent model 
of a damped oscillatory system that best describes the cross-line geophone 
coupling to the sea bed relative to that of a (theoretically) 
perfectly-coupled in-line sensor. The coupling response is then removed 
from the cross-line signals. It is assumed that the instrumental response 
characteristics are common to both receivers and are of no concern for 
purposes of this disclosure. 
At a first receiver station, a first common receiver gathers of seismic 
signal traces are assembled, from an in-line receiver, x.sub.s, and a 
second common receiver gather of seismic signal traces from a co-located 
cross-line receiver, y.sub.s. The signals derive from many source stations 
areally distributed over the three-dimensional volume of the region of 
interest. 
FIG. 6 is a panel showing first and second (counting from the left) in-line 
common trace gathers and third and fourth cross-line common trace gathers. 
The first and third and the second and fourth gathers are co-located. On 
the cross-line gathers, the first arrivals are greatly attenuated and a 
high-amplitude 20-Hz ringing-type interference is present. The 20-Hz 
interference is believed due to imperfect earth coupling. 
Source-receiver data pairs are selected having a source-receiver trajectory 
vector along an azimuth that is about 45.degree., .+-. some angular 
tolerance, to the mutual axial alignment of both of the 
horizontally-polarized receivers such as shown by the vectors 52 or 54, 
FIG. 4. Those data should therefore posses roughly equal signal levels in 
both components. It is preferable that a plurality of different 
source-receiver offsets, such as 0.0-500 meters be used over some 
preselected reflection-time window such as 3.0 seconds, counting from the 
first breaks. 
Auto correlate each in-line source-receiver trace pairs of the common 
receiver gather as follows: 
##EQU1## 
Similarly, auto-correlate the cross-line source-receiver trace pairs: 
##EQU2## 
where the in-line auto-correlations are normalized to unity and each 
cross-line auto correlations are normalized relative to its in-line 
companion. T is the length of the time window, .tau. is the phase lag and 
x.sub.s , y.sub.s are the trace-bin idents for the in-line and the 
cross-line traces. 
The cross-line responses must be balanced by a scale factor .chi. to 
correct for the level of the signal projected into the cross-line 
direction: 
EQU .chi.=.vertline.1/tan(.theta..sub.s -.theta..sub.x).vertline.(3) 
where .theta..sub.s is the source-receiver azimuth and .theta..sub.x is the 
orientation of the in-line receiver. The scaled auto-correlations 
.phi..sub.yy scaled by .chi. and the in-line auto-correlations 
.phi..sub.xx are then averaged .PHI..sub.xx and .PHI..sub.yy. .PHI..sub.xx 
represents an estimate of the source-receiver response, multiples of the 
geological response and earth attenuation. The .PHI..sub.yy response is 
representative of essentially the same parameters but with the cross-line 
coupling response added. 
The average response functions are transformed to the frequency domain to 
provide in-line and cross-line amplitude spectra as shown in FIG. 7 where 
the bold curve is the cross-line response. The spectral ratio, in the 
frequency domain, of the average cross-line response to the average 
in-line response is shown as the thin curve in FIG. 8. That curve was 
computed by deconvolving the cross-line response by the in-line response 
in the time domain. Specifically, the deconvolution operator is the 
inverse of .PHI..sub.xx, such that when it is convolved with .PHI..sub.xx, 
an impulse results from that operation. Convolving that operator with 
.PHI..sub.yy results in the thin curve of FIG. 8 after transformation to 
the frequency domain. If the average cross-line response, .PHI..sub.yy, 
were identical to .PHI..sub.xx, that response function would be an impulse 
with an otherwise flat response spectrum. The deconvolution could, of 
course by done on individual data pairs rather than on the average of the 
pairs, if desired. 
It is now required to determine the mechanical coupling parameters of a 
damped oscillatory system that best fits the observed spectrum. The 
parameters are the resonant or natural frequency .omega..sub.0 and damping 
parameter .eta. which can be determined by any number of well-known 
methods, one of which is presented here by way of example but not by way 
of limitation. The damped oscillatory system describing the coupling 
response may take the form 
##EQU3## 
where .omega. is the angular frequency and i=.sqroot.-1. It can be shown 
that 
##EQU4## 
where .OMEGA..sub.0 is the frequency at which the peak occurs in FIG. 8. 
Substituting (5) into (4) at the peak frequency where 
.omega.=.OMEGA..sub.0 and after a bit of algebraic manipulation, it can be 
shown that 
##EQU5## 
where .PHI..sub.yy (.OMEGA..sub.0) and .PHI..sub.yy (0) are the values of 
the frequency spectrum of the average auto-correlations after 
deconvolution at the maximum frequency and DC respectively. 
Equation (6) is solved iteratively for .eta. where the left hand side of 
(6) is greater than unity. Substituting .eta. in (5) gives .omega..sub.0. 
The bold curve in FIG. 8 is the best fitting damped oscillatory response 
for .omega..sub.0 and .eta.. 
The deconvolution operator is applied, by frequency domain division, to all 
of the cross-line geophone signal traces for a specific receiver station. 
FIG. 10 shows the results. The in-line signals are unchanged but the 20-Hz 
oscillatory response has been removed from the cross-line data panel. The 
phase effects of the cross-line coupling response as shown in FIG. 9 have 
been removed. The first-arrival transients on the cross-line panel have 
been enhanced. 
This invention has been described with a certain degree of specificity by 
way of example but not by way of limitation. Those skilled in the art will 
devise obvious variations to the examples given herein but which will fall 
within the scope and spirit of this invention which is limited only by the 
appended claims.