The document summarizes key concepts and developments in seismic velocity analysis in transversely isotropic (TI) media over several decades, beginning with Thomsen's (1986) seminal work introducing parameters to characterize TI anisotropy. Subsequent work expanded on non-hyperbolic moveout (Tsvankin and Thomsen 1994), dipping reflectors (Tsvankin 1995), and velocity analysis techniques for TI media (Alkhalifah and Tsvankin 1995). Later contributions improved modeling of non-hyperbolic moveout using rational functions (Douma and Calvert 2006) and the generalized moveout approximation (Fomel and Stovas 2010). The document outlines the theoretical underpinnings of seismic analysis in anisot

1. Seismic velocity analysis in
VTI media, an overview
Kamal Aghazade
aghazade.kamal@ut.ac.ir

2. Moveout velocity approximation (Thomsen, 1986)
Nonhyperbolic reflection moveout in anisotropic media (Tsvankin and Thomsen 1994)
Normal moveout from dipping reflectors in anisotropic media (Tsvankin, 1995)
Velocity analysis for transversely isotropic media, (Alkhalifah and Tsvankin, 1995)
Velocity analysis using non-hyperbolic moveouts in transversely isotropic media (Alkhalifah, 1997)
Nonhyperbolic moveout analysis in VTI media using rational interpolation (Douma and Calvert, 2006)
Generalized moveout approximation (GMA) (Fomel and Stovas , 2010)
The modified generalized moveout approximation: a new parameter selection (Stovas and Fomel,
2016)

3. The concept of anisotropy
Physical
properties look
the same in all
directions
Measurements
in any direction
have the same
result
Isotropic
material
Physical
properties are
not the same in
all directions
Measurements
in different
directions have
different results
Anisotropic
material
Transverse isotropy (TI)
Symmetric axis
transversely isotropic with a
Vertical symmetry axis (VTI)
transversely isotropic with a
Horizontal symmetry axis
(HTI)
transversely isotropic with a
Tilted symmetry axis (TTI)

4. 4
Isotropic processing of anisotropic world? Is it logical?

5. Thomsen’s approximations
I call it the Big Bang of Anisotropic processing of seismic data
Lets start

6. Equivalent to Helbig, 1983
Short spread Vnmo

7. 7
He measured the anisotropy from a comparison of the
well-log sonic data and the interval velocity profile obtained
from the surface seismic data and also from a comparison of
the seismic depth and the well-log depth and concluded that
depth anomaly in the North Sea basin is caused by the
velocity anisotropy of shale
By comparing layer thicknesses from
S-wave data with thicknesses from P-wave data. When the S-
wave thicknesses were significantly greater than
the P-wave (i.e., outside the range of expected errors), He
concluded that the layer was anisotropic.
He showed that anisotropy should be taken into account in
amplitude-offset studies involving shales.
They showed that for only one of the transversely isotropic
media considered here-shale-limestone-would v(z) DMO fail
to give an adequate correction within CMP gathers. For the
shale-limestone, fortuitously the constant-velocity DMO
gives a better moveout correction than does the v(z) DMO

8.  The standard hyperbolic approximation for reflection moveouts in
layered media is accurate only for relatively short spreads, even if the
layers are isotropic (Taner and Koehler, 1961).
 Velocity anisotropy may significantly enhance deviations from
hyperbolic moveout ( Liakhovitsky and Nevsky, 1971; Thomsen,
1986).
Hyperbolicassumption
analysis(anisotropicmedia)
Kery and Helbig
(1956)
Liakhovitsky and
Nevsky (1971)
Levin (1978,
1979, 1989)
Thomsen (1986)
Sheriff and
Seriram (1991)

Long
spread
studies
 Radovich and Levin (1982):
TI leads to spread-length-dependent
moveout velocity.
 Hake et al. (1984):
Suggested three-term Taylor series
expansion of 𝑡2 − 𝑥2 curves
 Berge (1991)
 Byun et al. (1989)
 Byun and Corrigan (1990).
Tsvankin and Thomsen 1994, Nonhyperbolic reflection moveout in anisotropic media

9. 9
Short-spread reflection moveout
Short-spread
limit
 for all three waves the short spread
moveout velocity is generally different
from the true vertical velocity.
 Winterstein (1986):
assumed small value of 𝛿 in order to estimate
𝛾.
 Banik (1984):
Found significant difference between vertical
P-wave velocities and moveout velocities in
North Sea shales.
 SV-wave may be more significantly
distorted by anisotropy than that for P-
wave.

10. 10
Elliptical anisotropy assumption
 wavefront is spherical for the SV-wave
and elliptical for P-wave
 all moveouts are strictly hyperbolic
(Levin, 1978).
 the elliptical anisotropy is a good
assumption for mathematical treatments.
However, the physics of real media says
𝜀 ≠ 𝛿.
Difference between the exact travel-
times and best-fit hyperbola
Tsvankin and Thomsen 1994, Nonhyperbolic reflection moveout in anisotropic media
We can introduce Deviation from
elliptical case (next slides)

11. 11
Intermediate-spread reflection moveout
Difference between the exact
travel-times and best-fit
hyperbola (Intermediate spread)
The Effective moveout velocity
of the best hyperbola normalized
by short spread moveout velocity
Tsvankin and Thomsen 1994, Nonhyperbolic reflection moveout in anisotropic media

12. 12
Intermediate-spread reflection moveout-weak
anisotropy approximation (WAA)
Tsvankin and Thomsen, 1994
Weak anisotropy approximation (Byun
et al, 1989 )
𝑡2
𝑧
𝑉𝛾
2
+
𝑥
2𝑉ℎ
2
𝑥2
𝑧2 +
𝑥
2
2
 require knowledge of vertical P-and S-
wave velocities.
Sena
(1991)
The WAA cannot explain some the pronounced deviation of the
SV-wave moveout from hyperbolic for 𝜎 < 0.

13. 13
On limitation of WAA
The second term may be
considered as the correction for
“strong” anisotropy.
WAA coefficients

14. 14
 For very small values of
𝛿 𝑎𝑛𝑑 𝜎, WAA breaks down.
 since usually 𝜎 > 𝛿, the WAA
is less suitable for the SV-wave
than for the P-wave.
Measure of inaccuracy of the conventional
hyperbolic moveout equation:

15. 15

16. 16
Long spread reflection moveout
 the three-term Taylor series provides
valuable results into peculiarities of non-
hyperbolic moveouts.
 the three-term Taylor series loses
accuracy rapidly with increasing offset.
Three-term Taylor series 𝑡 𝑇 and
approximation 𝑡 𝐴 for P-wave moveout
Considering large offset to depth ratio
(large incidence angles)

17. 17
Normal moveout from dipping reflectors in anisotropic media (Tsvankin, 1995)
Coincides with Thomsen’s (1986)
Agrees with Byun (1982) and Uren et al. (1990b)

18. 18
Comparison with Byun, 1984
P-wave moveout velocity calculated by Tsvankin formula
(dotted curve) and from travel-times (solid curve) for the
limestone-sandstone model from Byun (1984) .
P-wave moveout velocity calculated by Byun, 1984

19. 19
How many parameters determine the P-wave DMO signature?
𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑉𝑝0
𝑉𝑠0
, 𝜀 , 𝛿
𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑉𝑝0 , 𝜀 , 𝛿
The effect of 𝑉𝑠0The effect of 𝑉𝑝0
In VTI media the dip dependence of the P-wave NMO velocity is
primarily a function of 𝜀 𝑎𝑛𝑑 𝛿.

20. 20
The weak-anisotropy expression for the P-wave normal-moveout
velocity is sufficiently accurate for common small and moderate
values of 𝜀 and 𝛿.
Dip dependence of the P-wave moveout velocity for Vnmo
media is a function of only two parameters-𝜀 and 𝛿, with the
influence of the S-wave vertical velocity being practically
negligible.
The P-wave DMO signature is controlled, to a significant degree, by
the difference 𝜀 − 𝛿
If 𝜀 − 𝛿 < 0.15 − 0.2 non-hyperbolic moveout does not
seriously distort the P-wave moveout velocity on conventional
length spreads, even for steep reflectors.
This conclusion is the base
of many alternative
approaches.

21. 21
Velocity analysis for transversely isotropic media, Alkhalifah and Tsvankin, 1995.
 The main difficulty in extending seismic processing
to anisotropic media is the recovery of anisotropic
velocity fields from surface reflection data.
 Analytic and numerical analysis performed by
Tsvankin(1995) shows that dip-dependence of P-
wave NMO velocities is mostly controlled by 𝜀 − 𝛿.
 Alkhalifah and Tsvankin (1995) extended the NMO velocity
relation derived by Tsvankin (1995) by including ray
parameter.
 they showed that P-wave NMO velocity for dipping
reflectors in homogenous VTI media depends just on
𝑉𝑁𝑀𝑂(0) and new parameter 𝜂.
A 3D plot of NMO velocity
as a function of 𝜀 and 𝛿

22. 22
Velocity analysis for transversely isotropic
media, Alkhalifah and Tsvankin, 1995.
Field data example
Time migrated seismic line (offshore Africa)
Constant velocity stacks
(without accounting anisotropy)
Constant velocity stacks (with
accounting anisotropy)

23. 23
 Dellinger et al. 1993, Anelliptic
approximation for TI media.
 first anelliptic approximation:
the variation in P-wave phase slowness is described
in terms of three parameters:
𝑣ℎ, 𝑣𝑣 𝑎𝑛𝑑 𝑠ℎ𝑜𝑟𝑡 𝑠𝑝𝑟𝑒𝑎𝑑 𝑉𝑁𝑀𝑂 .
 second anelliptic approximation: adds the
vertical moveout velocity as an additional free
parameters.
 Sayers, 1995, Anisotropic velocity analysis.
 examined the method of Dellinger and showed
that the method is suitable for TI media for small
anellipticity.
 he provided new approach based an expansion of
the inverse-squared group velocity in spherical
harmonics.
 for TI media with small anisotropy his method
reduces to the method of Byun et.al (1989).
 the method does, however, require the use of
borehole measurements such as sonic logs or
VSP measurements to constraints the vertical
velocity.

24. 24
Alkhalifah, 1997, Velocity analysis using non-hyperbolic moveouts in transversely
isotropic media.
 the method proposed by Tsvankin (1994) ,Alkhalifah and Tsvankin (1995)
works only when reflectors with at least two distinct dips are present, as long
as one of the dips is not close to 90 degree.
 Alkhalifah (1997) used non-hyperbolic moveout in order to estimate 𝜂 and
discussed about the sensitivity of the inversion to errors in the measured
errors.
 He also applied semblance analyses over non-hyperbolic trajectories to
estimate both 𝑉𝑁𝑀𝑂 𝑎𝑛𝑑 𝜂.
Tsvankin (1994)
Tsvankin
And Alkhalifah
(1995)
Now the velocity analysis and
inversion requires 2 parameter
searching.

25. 25
Alkhalifah, T., Velocity analysis using non-hyperbolic moveouts in transversely
isotropic media.
Three layer model, the
first layer is isotropic
Percent time error in moveout
(departure from the exact moveout)
Alkhalifah, 1997
Tsvankin and
Thomsen, 1995
Modified Hake
et al. 1984
Some notes:
I. The derived equations reduce to
isotropic case when 𝜂 = 0.
II. the value of 𝜂 𝑒𝑓𝑓 also can be used
to describe the departure from
hyperbolic moveouts caused by
inhomogeneity in isotropic layered
media.

26. 26
Alkhalifah, T., Velocity analysis using non-hyperbolic moveouts in transversely isotropic media.
Degree of non-hyperbolic based on 𝜂 𝑒𝑓𝑓??
Values of 𝜂 𝑒𝑓𝑓 in an isotropic medium with a constant
velocity gradient, as a function of zero-offset time 𝑡0, for three
values of velocity gradient a.
 one cannot distinguish between the amount
of non-hyperbolic moveout attributable to
anisotropy and that attributable to
inhomogeneity
 medium, with large vertical inhomogeneity,
were strictly isotropic (𝜀 = 0 and 𝛿 = 0 in each
layer), then apprximately 𝜂 𝑒𝑓𝑓 = 0.06.
 In the presence of anisotropy resulted in
𝜂 𝑒𝑓𝑓 = 0.19.

27. 27
Velocity analysis for transversely isotropic media, Alkhalifah and Tsvankin, 1995.
Velocity analysis for various offset to depth ratios (based on hyperbolic
moveout)
Velocity analysis panels for various offsets
a) 𝜂 = 0
b) 𝜂 = 0.1

28. 28
Velocity analysis for transversely isotropic media,
Alkhalifah and Tsvankin, 1995.

29. 29

30. 30
Douma and Calvert, 2006, Nonhyperbolic moveout analysis in VTI
media using rational interpolation.
 studied the accuracy of the nonhyperbolic moveout equation
of Alkhalifah and Tsvankin (1995).
 they used rational interpolation to approximate
nonhyperbolic moveout in a VTI media.
 They showed the their approximation has close accuracy to
that method proposed by Fomel (2014).

31. 31
Douma and Calvert, 2006, Nonhyperbolic moveout analysis in VTI
media using rational interpolation.
Semblance scans and moveout-corrected gathers
True model parameters
Estimated model parameters
Conventional
nonhyperbolic
moveout
Rational
interpolation
method

32. 32
Fomel and Stovas , 2010, Generalized moveout approximation (GMA),
GMA:
A Five parameter approximation based on two rays with two
offset (x=0, x=X).
The parameters of original GMA are the travel time,
the second-order travel-time derivative, and the fourth-order
travel-time derivative computed at a zero-offset ray and the
travel time and first-order travel-time derivative computed
from a reference ray.
Calculated based on zero
offset ray
Defined from a reference ray

33. 33
Fomel and Stovas , 2010, Generalized moveout approximation (GMA),
Relative absolute error of different
traveltime approximations as a function
of velocity contrast and offset/depth ratio
for the case of a linear velocity model.
Relative absolute error of different
traveltime approximations as a function
of velocity contrast and offset/depth ratio
for the case of circular reflector.

34. 34
Fomel and Stovas , 2010, Generalized moveout approximation (GMA),
Dots:
Exact moveout from ray tracing in the one-dimensional
anisotropic Marmousi model
solid lines:
different approximations

35. 35
 basic theory of anisotropic wave propagation
 Influence of anisotropy on point-source radiation
and AVO analysis
 Normal-moveout velocity in layered anisotropic
media
 Nonhyperbolic reflection moveout
 Reflection moveout of mode-converted waves
 P-wave time-domain signatures in transversely
isotropic media
 Velocity analysis and parameter estimation for VTI
media
 P-wave imaging for VTI media

36. 36
Stovas and Fomel, 2016, Modified GMA
Same as GMA, but changes the definition of
parameter A:
defined from the second derivative of travel
time with respect to offset from a reference ray.
The relative error in the
estimation of travel-time curve
The error in the estimation of travel-time
curve versus offset/depth ratio for the
stack of VTI layers.

38. 38

39. 39

40. Homogeneity does not imply isotropy
Homogenous but not isotropic Isotropic at origin but not
homogenous