The Goos-Hänchen shift of wide-angle seismic reflection wave
LIU FuPing1,2*, MENG XianJun3, XIAO JiaQi4, WANG AnLing1 & YANG ChangChun2
1 Beijing Institute of Graphic Communication, Beijing 102600, China; 2 Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China;
3 Shengli Geophysical Research Institute of China Petrochemical Corporation (SINOPEC), Dongying 257022, China; 4 CNPC Greatwall Drilling Company, Beijing 100176, China
Received March 9, 2011; accepted July 1, 2011; published online December 15, 2011
The partial derivative equations of Zoeppritz equations are established and the derivatives of each matrix entry with respect to wave vectors are derived in this paper. By solving the partial derivative equations we obtained the partial derivatives of seismic wave reflection coefficients with respect to wave vectors, and computed the Goos-Hänchen shift for reflected P- and VS-waves. By plotting the curves of Goos-Hänchen shift, we gained some new insight into the lateral shift of seismic reflection wave. The lateral shifts are very large for glancing wave or the wave of the incidence angle near the critical angle, meaning that the seis-mic wave propagates a long distance along the reflection interface before returning to the first medium. For the reflection waves of incidence angles away from the critical angle, the lateral shift is in the same order of magnitude as the wavelength. The lateral shift varies significantly with different reflection interfaces. For example, the reflected P-wave has a negative shift at the reflection interface between mudstone and sandstone. The reflected VS-wave has a large lateral shift at or near the criti-cal angle. The lateral shift of the reflected VS-wave tends to be zero when the incidence angle approaches 90°. These observa-tions suggest that Goos-Hänchen effect has a great influence on the reflection wave of wide-angles. The correction for the error caused by Goos-Hänchen effect, therefore, should be made before seismic data processing, such as the depth migration and the normal-moveout correction. With the theoretical foundation established in this paper, we can further study the correction of Goos-Hänchen effect for the reflection wave of large incidence angle.
Goos-Hänchen effect, lateral shift, P- and SV-wave, reflection wave of wide-angle, Zoeppritz equations, critical angle
Citation: Liu F P, Meng X J, Xiao J Q, et al. The Goos-Hänchen shift of wide-angle seismic reflection wave. Sci China Earth Sci, 2012, 55: 852–857, doi: 10.1007/s11430-011-4344-5
The beam center of total reflection light laterally shifts a distance (xs) from the incidence point. This phenomenon has been demonstrated experimentally by Goos and Hän-chen [1] and is referred to as Goos-Hänchen effect [1–8]. It is illustrated in Figure 1, where n1 and n2 are the refractive indexes of media 1 and 2, and i is the incidence angle. The Goos-Hänchen effect became a hot research topic ever since its discovery. The conception of lateral shift was rapidly introduced into other branches of physics [1], such as
acoustics, plasma, quantum mechanics and so on [1]. In recent years, negative Goos-Hänchen shifts have been
discovered [3–5], shown in Figure 2 [3–5]. When a light ray propagates to the incidence point O on the reflection inter-face, the reflection light ray laterally shifts in the negative direction of x axis. The reflection light propagates along the direction of AD, just as if the light is reflected at the point A. This phenomenon is called negative Goos-Hänchen shift, which has been proved with experiment and theoretical study [3–5]. The negative lateral shift has also been found with the total reflection SH-wave.
Liu F P, et al. Sci China Earth Sci May (2012) Vol.55 No.5 853
Figure 1 Lateral shift of total reflection light.
Figure 2 Negative shift of reflected light wave.
The lateral shift of Goos-Hänchen effect is in the same order of magnitude as the wavelength of seismic wave when the incidence is in some specific angles, though the lateral shift may be much larger than the wavelength [9–12]. In general, the longer the wavelength, the larger the lateral shift.
Large-angle reflection problems [13–23] are often en-countered in geophysics, such as cross-well seismic and the exploration of weak reflection layers under over-thrust faults or high-speed shielding layers. Lateral shifts always exist with those problems of seismic reflected wave. The results of seismic data processing are significantly affected by the lateral shift. Due to the lack of theoretical study, however, the lateral shift has not been taken into account in the methods of seismic data handling [24–33].
In recent years, Liu et al. [11, 12] have studied the lateral shift of incidence waves of SH- and SV-waves, where the reflection coefficients can be expressed explicitly. For com-plicated cases of P- and SV-conversion waves, however, conventional methods cannot be used to compute the lateral shift of the reflection waves because the analytical expres-sions of Zoeppritz equations are very complicated for them. We need to develop a new method to compute the lateral shift for the P- and SV-conversion waves.
Starting from the partial derivative equations of Zoep-pritz equations, in this paper, we will derive the derivatives of each matrix entry with respect to wave vector, describe a method for the computation of the lateral shift, and present computed Goos-Hänchen shifts for reflection P- and VS- waves. The theoretical foundation established in this paper is further used to study correction methods for Goos-Hän- chen effect of the wide-angle reflection waves.
1 Reflection coefficients of seismic waves
The reflection and refraction of P-wave at an interface be-tween two elastic media is shown in Figure 3, where p and sv designate P- and SV-waves, respectively; is the inci-dence angle of the P-wave; is the reflection angle of the reflected SV-wave; ′ and ′ are the refraction angles of the P- and SV-waves; vp1, vs1, vp2, and vs2 are the propaga-tion velocities of the P- and S-waves in medium 1 and me-dium 2; 1 and 2 are the densities of medium 1 and medi-
um 2; k1p, k1s, k2p and sk2 are the wave vectors of P- and
S-waves. At the interface between two elastic media, Zoep-pritz equations can be written as [26, 27]:
AR = B, (1)
where
T, , , ,PP PS PP PSR R R T T
T21
1
sin ,cos , cos2 , sin 2 ,s
p
vB
v
2 21 2 2
1 1 1 1 1
2 21 2 2 2 2
11 1 2 1
sin cos sin cos
cos sin cos sin
cos2 sin 2 cos2 sin 2 .
sin 2 cos2 sin 2 cos2
ps s
p p p
s s ss
p p
A
vv v
v v v
v v vv
v v
(2)
RPP, RPS, TPP and TPS are the reflection and refraction coef-ficients of a P-wave at an interface between two elastic me-dia, respectively.
Figure 3 Reflection and refraction of a P-wave at an interface between two elastic media.
854 Liu F P, et al. Sci China Earth Sci May (2012) Vol.55 No.5
2 Computation method of Goos-Hänchen shift of seismic wave
Denoting kpx and ksx to be the x-components of wave vectors of P- and S-waves, for simplicity, kpx and ksx are expressed in the form of k, where ={p, s}. The partial derivatives of Zoeppritz equations with respect to k can be expressed as
.R A B
A Rk k k
(3)
Equation (3) is the partial derivative equations with respect to wave vectors. The reflection coefficients R are obtained by solving Zoeppritz equations (1). Once /A k and
/B k are there, the partial derivatives with respect to k
can be obtained by solving eq. (3). Setting
,R i
, , , ,k k k
RR i i
k k k
(4)
the phase shift of reflection coefficients can be defined as 1tan ( / ) , where i is an imaginary unit, and are
the real and imaginary parts of reflection coefficients. The lateral shift of a seismic wave can be computed as
1
2
2 2
tan
.
xk k
k k
(5)
With the lateral shift x, the time delay of reflected seismic wave caused by Goos-Hanchen effect can be calculated with formula:
/ .t x v
3 Matrix element of the partial derivative equations of wave vector
With the geometrical relations of wave vectors (k1x=ksx=kpx), we have
1
1
sin ,xk
k
2
1
1
cos 1 ,xk
k
1
1
,p
kv
1x
A A
k k
and 1
,x
B B
k k
where sin and cos are functions of k1x.
The partial derivatives of sin and cos with respect to k can be expressed as
11
1 1 1 1
sin 1,px
x x
vk
k k k k
(6)
2 2
21 11 1
11 1 1
cos1 ( / ) 1x x
xx x
k kk k
kk k k
1
21
sintan .
1 sin
pv
k
(7)
According to Snell’s formulas [26] 1 1/ sin / sinp sv v
2 2/ sin / sinp sv v and the relations of trigonomet-
ric functions, we have 1 1sin ( / )sin ,s pv v sin
2 1( / )sin ,p pv v 2 1sin ( / )sin ,s pv v
cos 2
1
1
1 sin ,s
p
v
v
2
2
1
cos 1 sin ,p
p
v
v
cos 2
2
1
1 sin .s
p
v
v
Similarly,
1 1 1
1 1 1 1 1
sin sin,s s s
x p x p
v v v
k v k k v
1
1
costan ;s
x
v
k
2 2 2
1 1 1 1 1
sin sin,p p p
x p x p
v v v
k v k k v
2
1
costan ;p
x
v
k
2
1
sin,s
x
v
k
2
1
costan ;s
x
v
k
1 1 1
1 2
sin 2 sin cos2 cos 2sin
2 cos (1 tan ),
x x x
p
k k k
v
2
1 1
1
1
cos21 2sin
4sin4sin sin ,
x x
p
x
k k
v
k
21
1
sin 22 cos (1 tan ),s
x
v
k
1
1
4cos2sin ,s
x
v
k
2 2
1
sin 22 cos (1 tan ),p
x
v
k
Liu F P, et al. Sci China Earth Sci May (2012) Vol.55 No.5 855
2
1
4cos2sin ,p
x
v
k
22
1
sin 22 cos (1 tan ),s
x
v
k
2
1
4cos2sin .s
x
v
k
Substituting above partial derivatives of trigonometric func-tions into 1/ xA k and 1/ ,xB k we have
1
1
x x sx px
A A A A
k k k k
1 1 2 2
1 1 2 2
2 22 2 21 2 2
11 1 1 1 1
2 22 2 2 2 2 21 1
1 1
tan tan
tan tan
42 2.4 sin ( ) sin ( )
2 ( ) 4 sin 2 ( ) 4 sin
p s p s
p s p s
p ss ss
p p p
s ss s
v v v v
v v v v
v vv vv f f
v v v
v vv f v f
(8)
21 1 1 1
1
1tan 4 sin 2 ( ) ,p p s s
x x
B Bv v v v f
k k
(9)
where 2( ) cos (1 tan ).f x x x
4 Numerical examples of lateral shifts for re-flection wave of wide-angle
In order to fully understand the characteristics of Goos- Hänchen effect, we have calculated the lateral shifts for a few typical reflection interfaces, which have been met in prospecting and exploration of oil and gas. Figure 4 is the lateral shift curves of seismic reflected wave on a reflection interface between mudstone and sandstone. Where, the fre-
quencies of waves f={35, 50} Hz; the elastic parameters are
In this numerical example, the contrast of elastic param-eters between the two media is relatively large, there are two critical angles: the first critical-angle c1=27.90 and the second critical-angle c2=51.5°.
From Figure 4 we can find that for glancing wave or the wave that the incidence angle is near the two critical angles, the lateral shifts are very large, especially, at the First criti-cal-angle and =90°. The lateral shift curves of P-wave are the discontinuous. When the incidence angle is between the First critical-angle and the Second critical-angle, the lateral shifts of P-wave are about a few meters (Figure 4(a), (b)). The lateral shift of SV-wave is in the range between 10 to 20 m (Figure 4(c)). When the incidence angle is larger than the second critical-angle, there are negative points on the lateral shift curves of both P- and S-waves, indicating nega-tive shifts.
Figure 5 is the lateral shift curves of seismic reflected wave at the interface between mudstone and tight sand, where, elastic parameters are
Mudstone: vp1=3850 m/s, vs1=2300 m/s, 1=2650 kg/m3. Tight sand: vp2=5500 m/s, vs2=3200 m/s, 2=2700 kg/m3. The frequencies of waves f={35, 50}Hz. First critical-
angle c1=44.5° and there is no Second critical-angle. This figure shows that the lateral shift of P-wave is about 50 m with most incidence angles (Figure 5(a)). There is a maxi-mum point on the lateral shift curve of the S-wave (Figure 5(b)). The lateral shift of the S-wave tends to be zero as the incidence angle approaches 90°.
When the reflection happens at interface between a gas-saturated sandstone (vp1=2190 m/s, 1=2000 kg/m3, vs1=1460 m/s) and an oil-saturated sandstone (vp2=2440 m/s, 2=2250 kg/m3, vs2=1172 m/s), the lateral shift curves are shown in Figure 6 (c1=63.8°, f=35 and 50 Hz). The lateral shift curve of P-wave has two discontinuous points at the First critical-angle and =90° (Figure 6(a)). The lateral shift of the S-wave decreases with the increase of incidence an-
Figure 4 Lateral shift of seismic reflected wave at the interface between mudstone and sandstone. (a) Lateral shift of reflected P-wave; (b) magnified lat-eral shift of reflected P-wave; (c) lateral shift of reflected SV-wave.
856 Liu F P, et al. Sci China Earth Sci May (2012) Vol.55 No.5
gle (Figure 6(b), becoming zero when reaches 90°. Figure 7 shows the lateral shift curves of seismic reflect-
ed wave at an interface between oil-saturated sandstone (vp1=2440 m/s, 1=2250 kg/m3, vs1=1172 m/s) and wa-ter-saturated sandstone (vp2=2660 m/s, 2=2300 kg/m3, vs2=1278 m/s; c1=668°, f=35 and 50 Hz). The primary characteristics of curves are the same as in Figure 6, with a different magnitude of the lateral shift.
5 Conclusions
We have established the partial derivative equations of Zo-eppritz equation, developed a method for the computation of the lateral shift of Goos-Hänchen effect, and accurately computed lateral shifts for reflection P- and VS-waves.
Numerical examples reveal some facts about Goos- Hänchen effect of reflected seismic waves. The lateral shifts
Figure 5 Lateral shift of seismic reflected wave at interface between sandstone and tight sand. (a) Lateral shift of reflected P-wave; (b) lateral shift of re-flected S-wave.
Figure 6 Lateral shift of seismic reflected wave at the interface between gas-saturated sandstone and oil-saturated sandstone. (a) Lateral shift of reflected P-wave; (b) lateral shift of reflected S-wave.
Figure 7 Lateral shift of seismic reflected wave at interface between oil-saturated sandstone and water-saturated sandstone. (a) Lateral shift of reflected P-wave; (b) lateral shift of reflected S-wave.
Liu F P, et al. Sci China Earth Sci May (2012) Vol.55 No.5 857
of P-wave are very large for glancing wave or the wave of incidence angle near the critical angle. With incidence angle other than critical angles, the lateral shift is in the same or-der of magnitude as the wavelength. We have also discov-ered that the lateral shift can be negative for both P- and S-waves.
The results of seismic data processing of wide-angle re-flection waves are severely influenced by the Goos- Hänchen effect. In exploration practice, because the wide-angle reflection wave can form a clear seismic section, and have a strong reflection amplitude, the problems of wide-angle reflection have been taken into account. How-ever, there is Goos-Hänchen effect where is total reflection.
The quantity of lateral shift can be tens of meters even hundreds of meters, significantly affecting the results of seismic data processing. In order to improve the precision of seismic interpretation, the Goos-Hänchen effect of wide- angle reflection wave should be corrected for in the seismic data processing.
The theory and computation method presented in this paper can be applied to analyze the influence of lateral shift on the reflected P- and VS-waves, and to research for the correction of Goos-Hänchen effect in the seismic depth mi-gration and the normal-moveout of large-angle reflection wave.
This work was supported by Funding Project for Academic Human Re-sources Development in Institutions of Higher Learning (Grant No. PHR201107145).
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