Rapid estimation of converted wave velocities using prestack migration
John C. Bancroft, Thais Guirigay
Helen Isaac
30 November, 2012
1
0 500 1000 1500 2000 2500 30001
1.5
2
2.5
3
3.5
4
4.5
5
Depth (m)
Gam
ma γ
Gamma functions in depth from picked velocities
γ: Int. Vels. γ: RMS vels. γ: log
Motivation
2
Outline • Objectives • Review EOM P-P and P-S, Vc • Estimating Vc • Estimation Vs • Formation of Gathers • Comments and Conclusion • Results next talk
4
Scatter point
SP R S MP
t/2
t0/2 or z0 hyperbola
x h h
2te
( ) ( ) 2 20
2
2 22 20 0
2 24 42
4ex h x ht tt h
VVt
V+ −
= + + + +=
E
Kirchhoff Prestack Migration
5
Equivalent Offset Migration
Scatter point
SP R S MP
t/2
t0/2 or z0 hyperbola
he
x h h
2te
( ) ( ) 2 20
2
2 22 20 0
2 24 42
4ex h x ht tt h
VVt
V+ −
= + + + +=
2 22 2 2
2 2
4eh x hx h
t V= + −
E
6
• Approximate Vp to form gathers
• More accurate Vp obtained from gathers
• For then m/s
• Not the case for converted wave data
• Need a more accurate starting velocity Vc
10
The point …
1Vp = ∞ 2 3200Vp =
Converted waves
11
P-P Source Receiver CMP
Z P
P
θ θ
P-S Source Receiver CMP
CCP
θ φ
Z P S
S R
Conversion Points
Asymptotic Approximation
Midpoint
Xp
2h
Most important up shallow X
Equivalent offsets: converted waves hr
hs
2 20 0
2
2 22 20 0
2 2
2
2
2
4 4 4 4p ps r
rms P rms P
s s
rm S
e
s r
e
ms S
ht th ht tV V V
htV
− −
− − −
− −
−
= + + + = + + +
12
Scatter point
SP R S MP
t/2
hyperbola
he
2te
E
t0p, t0s, z0
Vrms and Vave
0 0.5 1 1.5 21500
2000
2500
3000
3500
4000
4500
Time T0 (sec)
Vel
ocity
(m/s
)Comparison of Vp-rms, Vp-int, and Vp-ave.
Vp-intVp-rmsVp-ave
RMS velocity slightly higher than average velocity. 13
Vrms / Vave
0 0.5 1 1.5 20
0.5
1
1.5
Time T0 (sec)
ratio
Ratio Vp-rms/Vp-ave
P
0 0.5 1 1.5 20
0.5
1
1.5
Time T0 (sec)
ratio
Ratio Vp-rms/Vp-ave over Vs-rms/Vs-ave
P/S
0 0.5 1 1.5 20
0.5
1
1.5
Time T0 (sec)
ratio
Ratio Vs-rms/Vs-ave
S
14
2 2 2 2 20
2 2 20 00
1 1 1 ˆˆ ˆ ˆ1s r
rms p rms pe
rms s rme
s s
z hV V
t z h z hV
hV
z− −− −
= + + + +++=
2 20
2 ˆ ec
ztV
h+=
Assume rms p rms s
ave p ave s
V VkV V
− −
− −
≈ ≈
Define 0 0ˆ rms
ave
Vz zV
=
2220ˆ4
ceh t V z= −
2 rms p rms sc
rms p rms s
V VV
V V− −
− −
=+
15
Converted wave velocity Vc
• Initial Vc1 with Vp and • Limited range EO gathers • Pick new Vc2
• Estimate Vs
• Full EO gathers with Vp and Vs • Pick new Vc3
• Moveout correction with Vc3 • Stack to complete the prestack migration
2 rms p rms sc
rms p rms s
V VV
V V− −
− −
=+
γ
16
1. Using
2. Narrow range gather
Initial Vc can be formed
Estimating of Vc1
,p
s
VV
γ = 2γ =
maxx x
12(1 )
rms Pc
VVγ−=
+
( ) ( )2 22 20 0
1 1ˆ ˆrm rms Ss P
t z x h zV
x hV −−
= + + + + −
( ) ( )2 22 20 0 0
1 1ˆ ˆrms C m C
xr s
t z x hV V
z x h−
→−
≈ + + + + −
17
Form few gathers using one velocity Vc1 Velocity analysis to find Vc2 Now find Vs
Estimating Vs from Vc2
2 rms S
rms
rms Prms C
r s P Sm
VVV V
V −−−
− −
=+
2rms C rms
rmP
rms P rs S
ms C
V VV
VV
−
− −−
−=−
20
2 20
2 ˆrms C
p s et hV
z−
− = +
2 2 2 20 0
1 1ˆ ˆ( ) ( )rmsm S
p sr s P
t z x h z x hV V −
−−
= + + + + −
Full CCSP gathers
Use he to form CCSP gathers
21
• is defined with Vint and depth
• Vp, Vs, and Vc are also defined in depth
• Using map depth to tc
• Keep all times in tc
We have ignored t times 2 ( ) ( )
( ) ( )srms P rms S
rms Cr
p
pP sms rms S
tV VV
tV Vt t
− −−
− −
=+
Time t consideration
γ
Ave RMSV V≈
z
zz
•
z
23
Matching the traveltime for P- and C-wave data
Method 1 1. Convert Vrms-p(top) to interval Vint-p (top)
2. Use interval velocities to map the times to depth
3. Get average velocity Vave-p of Vrms-p
0opt z⇒
0 0.5 1 1.5 2 2.5 3 3.5 42000
3000
4000
5000
6000
Time T0 (sec)
Vel
ocity
(m/s
)
Comparison of Vp-rms, Vp-int, and Vp-ave.
Vp-rmsVp-intVp-ave
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Method 1 (continued) 4. Scale the amplitude Vint-c to Vint-p at z (same as top)
using 4. Use Vint-c (z) and the corresponding depth
increments, compute the C time at each depth.
5. Resample Vint-c from irregular times to equal time increments
6. Convert the interval C velocities to RMS C velocities
7. Get the depth of every Vc time sample using Vint-c
Matching the traveltime for P- and C-wave data
λ
25
Matching the traveltime for P- and C-wave data
Method 2 1. Using the corresponding average velocities
2. Resample Vrms-c(n) to Vrms-c(m) using equal increments of m
2 ( )(1 ( ))
rms prms c
V zV
zγ−
− =+
12oc opt t γ+
≈ 12
m nλ+=
26
0 1 2 3 4 5 61500
2000
2500
3000
Time T0 (sec)
Vel
ocity
(m/s
)
Comparison of Vc-rms1 and Vc-rms2
Vc-rms1Vc-rms2
Estimating the C velocities
27
0 2000 4000 6000 8000 100000
1000
2000
3000
4000
5000
6000
Depth (m)
Vel
ocity
(m/s
)
Comparison of interval velocities in depth
Vp-int(z)Vc-int(Gz)Vc-int(Pz)
Estimating the C velocities
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Method 1 1. From the Vc picked:
2. Convert shear interval Vint-s from depth to time
3. Convert Vint-s in S time into Vrms-s
4. Convert tos to top
5. Get the RMS gamma function in time
6. Convert RMS gamma to depth assuming Vrms = Vave
int intint
int int
( )2
p cs
p c
V VV z
V V− −
−− −
=−
2int
00
0
( )( )
N
s nn
rms s s N
nn
V n tV t
t
−=
−
=
=∑
∑
Estimating the S velocities
( )( )
( )rms p p
rmsrms s p
V tt
V tλ −
−
=
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0 1 2 3 4 50
1000
2000
3000
4000
5000
6000
Time T0 (sec)
Vel
ocity
(m/s
)
Comparison of interval velocities in time
Vp-intVc-int(P)Vs-int(P)
Estimating the S velocities
30
0 2000 4000 6000 8000 100000
1000
2000
3000
4000
5000
6000
Depth (m)
Vel
ocity
(m/s
)
Comparison of interval velocities in depth
Vp-int(z)Vc-int(Pz)Vs-int(Pz)
Estimating the S velocities
31
0 2000 4000 6000 8000 100001
1.5
2
2.5
3
3.5
4
4.5
Depth (m)
Gam
ma
Comparison of interval Gamma functions
Gamma from Vp-rms(Gz)Gamma from picked Vc-rms(Pz)Gamma from well-log
Estimating the S velocities
32
Another project in N.E. BC
0 500 1000 1500 2000 2500 30001
1.5
2
2.5
3
3.5
4
4.5
5
Depth (m)
Gam
ma γ
Gamma functions in depth from picked velocities
γ: Int. Vels. γ: RMS vels. γ: log
33
• Use EO concepts to process P-S data • Need approximate velocity starting velocities
Vp and Vs • Can use a single velocity Vc most of the time • Simple process to estimate of Vc and Vs
– Vc1 from Vp and – Vc2 from initial gather – Vs from Vp and Vc2
• Produced reasonable values for
Comments and conclusions
γ
35
γ
γ
• Use EO concepts to process P-S data • Need approximate velocity starting velocities
Vp and Vs • Can use a single velocity Vc most of the time • Simple process to estimate of Vc and Vs
– Vc1 from Vp and – Vc2 from initial gather – Vs from Vp and Vc2
• Produced reasonable values for
Comments and conclusions
γ
36
γ
γ
• Use EO concepts to process P-S data • Need approximate velocity starting velocities
Vp and Vs • Can use a single velocity Vc most of the time • Simple process to estimate of Vc and Vs
– Vc1 from Vp and – Vc2 from initial gather – Vs from Vp and Vc2
• Produced reasonable values for
Comments and conclusions
γ
37
γ
γ
• Use EO concepts to process P-S data • Need approximate velocity starting velocities
Vp and Vs • Can use a single velocity Vc most of the time • Simple process to estimate of Vc and Vs
– Vc1 from Vp and – Vc2 from initial gather – Vs from Vp and Vc2
• Produced reasonable values for
Comments and conclusions
γ
38
γ
γ
• Use EO concepts to process P-S data • Need approximate velocity starting velocities
Vp and Vs • Can use a single velocity Vc most of the time • Simple process to estimate of Vc and Vs
– Vc1 from Vp and – Vc2 from initial gather – Vs from Vp and Vc2
• Produced reasonable values for
Comments and conclusions
γ
39
γ
γ
• Use EO concepts to process P-S data • Need approximate velocity starting velocities
Vp and Vs • Can use a single velocity Vc most of the time • Simple process to estimate of Vc and Vs
– Vc1 from Vp and – Vc2 from initial gather – Vs from Vp and Vc2
• Produced reasonable values for
Comments and conclusions
γ
40
γ
γ
• Use EO concepts to process P-S data • Need approximate velocity starting velocities
Vp and Vs • Can use a single velocity Vc most of the time • Simple process to estimate of Vc and Vs
– Vc1 from Vp and – Vc2 from initial gather – Vs from Vp and Vc2
• Produced reasonable values for • Can approximate Vave with Vrms
Comments and conclusions
γ
41
γ
γ