# Gaiser’s coupling analysis of geophone data

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

Variations in geophone coupling contaminate signal amplitudes registered by the geophone components, and need to be compensated for in a surface-consistent manner. Because of coupling problems, what is recorded by each one of the three geophones is not exactly the same as the ground motion at the seabed. A frequency-domain model equation that relates the recorded signal components {X′(ω), Y′(ω), Z′(ω)} by the three geophones in the inline, crossline, and vertical directions (x, y, z), respectively, and the actual ground motions in the three orthogonal directions {X(ω), Y(ω), Z(ω)} is given by 

 ${\begin{pmatrix}X^{\prime }(\omega )\\Y^{\prime }(\omega )\\Z^{\prime }(\omega )\\\end{pmatrix}}={\begin{pmatrix}I&0&0\\0&C_{y}&C_{z}\\0&V_{y}&V_{z}\end{pmatrix}}{\begin{pmatrix}X(\omega )\\Y(\omega )\\Z(\omega )\\\end{pmatrix}},$ (66)

where ω is the angular frequency, I is unity, and the nonzero elements Cy, Cz, Vy, and Vz describe the coupling response of the geophones.

Note from equation (66) that X′(ω) = X(ω); this means that we assume that the inline geophone is perfectly coupled. Since the inline geophone is guided by the cable itself, this is considered a valid assumption in practice. Whereas the vertical and crossline geophones are not coupled completely — hence the nonzero elements of the coupling matrix. The imperfect coupling leads to vertical and crossline geophone signals mutually contaminating each other in a manner that can be modeled by equation (66).

We wish to estimate the ground motion vector {X(ω), Y(ω), Z(ω)}; this requires inverting equation (66) as given by Gaiser 

 ${\begin{pmatrix}X(\omega )\\Y(\omega )\\Z(\omega )\\\end{pmatrix}}={\frac {1}{D}}{\begin{pmatrix}D&0&0\\0&V_{z}&-C_{z}\\0&-V_{y}&C_{y}\end{pmatrix}}{\begin{pmatrix}X^{\prime }(\omega )\\Y^{\prime }(\omega )\\Z^{\prime }(\omega )\\\end{pmatrix}},$ (67)

where D = VzCy − VyCz and is the determinant of the coupling matrix in equation (66).

From the matrix equation (67), write explicitly the recovered ground motions

 $Y(\omega )={\frac {V_{z}}{D}}Y^{\prime }(\omega )-{\frac {C_{z}}{D}}Z^{\prime }(\omega )$ (68a)

and

 $Z(\omega )=-{\frac {V_{y}}{D}}Y^{\prime }(\omega )+{\frac {C_{y}}{D}}Z^{\prime }(\omega ).$ (68b)

The coupling compensation operators are estimated in a surface-consistent manner  with the constraint that, following the rotation, the energy of the transverse component is minimum.

Gaiser  reported a coupling experiment to verify the validity of the coupling theory described above. Figure 11.6-12a,b,c show the inline, crossline, and vertical geophone records obtained from an OBC survey. In the same figure, the record associated with the crossline geophone is shown after compenating for coupling (Figure 11.6-12d). To study the validity of the compensation based on equation (68), a diver firmly planted the receiver unit into the seabed and the recording was repeated. The resulting crossline record is shown in Figure 11.6-12e. If the coupling theory holds, then the records in Figures 11.6-12d,e should look very similar. Differences may be attributed to poor coupling of the planted receiver unit.

Figure 11.6-13 shows the result of surface-consistent coupling analysis. To apply the coupling corrections, scale the amplitudes in a given geophone record by the product of the source scalar and the associated component scalar. Figure 11.6-14 shows the common-shot gather as in Figure 11.6-6 after the application of surface-consistent amplitude corrections (Figure 11.6-12). The same shot gather with AGC is shown in Figure 11.6-15. The common-receiver gather as in Figure 11.6-8 after the application of surface-consistent amplitude corrections (Figure 11.6-12) is shown in Figure 11.6-16. The same receiver gather with AGC is shown in Figure 11.6-17. To examine the degree of compensation for differences in geophone coupling, refer to the close-up displays shown in Figures 11.6-18 and 11.6-19.