In this presentation I tried to talk about Thomesn parameters based on published papers and related documents.

1. Thomsen Parameters and
their constraints for TI media
Kamal Aghazade

2. Part1:
Thomsen parameters
Part 2:
 physical constraints on Thomsen parameter ( )𝛿

3. Part 1:
Introducing Thomsen parameters to seismic
anisotropy
3

4. Seismic isotropy vs anisotropy
Isotropy comes from the Greek words isos (equal)
and tropos (way) and means uniform in all
directions. Isotropic materials like glass exhibit the
same material properties in all directions.
Physical properties are
not direction dependent.
Anisotropy: physical properties
are direction dependent.
Thomsen,
2014
Seismic anisotropy:
The dependent of seismic velocity
upon angle.
4

5. Beyond the simple definition of seismic anisotropy
a) isotropic, b) anisotropic media
Shear wave splitting in anisotropic media
https://en.wikipedia.org/wiki/Shear_wave_splitting
5

6. Beyond the simple definition of seismic anisotropy
isotropic wave equation simulation
6
anisotropic wave equation simulation

7. Lets go further: mathematical description Elastic media
Linear Elastic material and Hook’s law
7

8. seismic anisotropy media
(Transverse Isotropy)
Common coring configuration for anisotropy
measurements. Meléndez-Martínez (2014).
Rüger, 1997
http://www.glossary.oilfield.slb.com
8

9. Stiffness matrix in Transverse Isotropy 9

10. Christoffel equation
(eigenvalue
problem)
Directional dependence of three phase velocities
(Daley and Horn, 1977)
Algebraically complex 
10Transverse Isotropy

11. A simpler and meaningful structure for anisotropic analysis (Tsvankin, 2012).
Continue on next page 
11

12. The problems mentioned above are already solved by Thomsen (1986)
12

13. Weak elastic anisotropy and “anisotropies”
We can recast the equation for P-SV and SH waves velocity by
using notation involving:
1 – two elastic moduli (Vertical P- and S-waves velocity )
+
2- Three measures of anisotropy (We call them anisotropies).
131 – Simplifying Velocity equations for
types of waves
2 – anisotropies are non-dimensional.
So, one may speak of X percent
anisotropy.
3 – reduce to zero in the degenerate
case of isotropy. So, material with small
value of anisotropy may be denoted:
“Weakly anisotropic”

14. 14
the algebraic complexity of new equations impedes a clear
understanding of their physical content. Progress may be
made. however, by observing that most
rocks are only weakly anisotropic. even though many of
their constituent minerals are highly anisotropic (Thomsen,
1986).
Thomsen’s revolution
14

15. Weak anisotropy
Based on laboratory data Thomsen (1986) showed
that most of evaluated rocks have anisotropy in the
weak to moderate (i.e. less than 0.2 ).
We use Taylor series to expand equations for
anisotropies and just retain only linear terms.
Now, we can talk about anisotropies
based on angles. 
15

16. Some notes about Thomesn parameters 16

17. By measurements at 0, 45 and 90 degrees we
have:
Lets consider error propagation in 𝛿 17

18. For more information about previous topics see:
Thomsen, L., [2014], Understanding Seismic Anisotropy
in Exploration and Exploitation, Second edition, SEG.
Tsvankin, I.,[2012] Seismic Signatures and Analysis of
Reflection Data in Anisotropic Media
,Third Edition, Society of exploration Geophysicists.
And related references.
Leon Thomsen
Ilya Tsvankin

19. Part 2:
physical constraints on 𝐶13 & 𝛿
for
transversely isotropic hydrocarbon source rocks
Yan et.al (2015)

20. 20

21. 21

22. 22

23. Coordinate system for
following relations
Young’s modulus and Poisson’s ration for TI media (see King,1964) 23

24. Schema of deformation of vertical plug
(left) and horizontal plug (right)
of organic shale under axial
compressional testing. Dark grey
represents plugs
before deformation, and light grey
represents plugs after deformation
24

25. 1- measurements
uncertainty
2 – material should not
be classified as a TI
medium
If a TI medium is
infinitely stronger in
the horizontal direction
compared with vertical
direction.
𝜐 𝐻𝑉 → 1
25

26. 26
Setting constraint on 𝐶13

27. 27On behavior of 𝛿

28. We have a constraint on 𝐶13
Substituting in 𝛿
Rewriting the
relation based on
other Thomsen’s
parameters
28

29. Where
29

30. Constraints on anellipticity parameter
“anellipticity”
parameter (η) that describes the degree of deviation from
elliptic anisotropy.
The anellipticity parameter η is important for anisotropic
seismic data processing because it determines the relation
between the normal moveout velocity and the horizontal
velocity (Tsvankin 2012).
30
Lower and upper bounds for 𝜂

31. Laboratory data and the constraints 31
several data points have negative values. The
corresponding 𝛿 values are above the high bound,
and they tend to have higher values of δ.
𝜐 𝐻𝐻
here are quite a few points with 𝜐 𝐻𝐻 > 𝜐 𝐻𝑉 . The
corresponding values are lower than the low bound,
and they tend to have lower values of δ .
𝐶13
About two-thirds of the data points lie in the center area,
where believed that all the hydrocarbon source rocks with
TI anisotropy should lie within.
There is some uncertainty in the middle part of the figure.
there are more data points lying below the low bound than
above the high bound.

32. Uncertainty in labratory velocity anisotropy measurements
Laboratory velocity anisotropy measurement on TI media
requires at least five velocity component measurements,
among which one velocity measurement must be made in
oblique direction.
32

33. Case I: Negative angle error
make about 20% of the data points lie
below the low bound
2 𝑜
Case II: Negative 5 𝑜
angle error
make about 62% of the data points
lie below the low bound
Case III: Positive 5 𝑜 angle error
make
about less than 8% of the data points lie above
the high bound
1) If the phase velocity in 45 𝑜
is
underestimated by 1%, 22% of the data
points move below the low bound.
2) If the phase velocity in 45 𝑜
is
overestimated by 1%, only one data
point moves above the high bound
33

34. difference between group and phase velocities
ray tracing of ultrasonic velocity measurement on the 45ºplug (left),
transmission time versus angle (right)
34
I. if the transducer is not wide enough (or
the sample is too long), the first
arriving energy might be missed by the
receiving transducer and the phase
velocity tends to be underestimated

35. Phase to group correction effect 35

36. Applications
The effect of the other Thomsen parameters on δ .
36

37. 37
Histogram of from
laboratory velocity anisotropy measurement.
𝐶11−2𝐶66
here are only 2 data points with
One data point is due to data entry error and the other data
point is due to signals of substandard quality.
𝐶11−2𝐶66 < 0
37

38. The trends of the approximated bounds comply well with the laboratory
measured data if data points lying outside of the δ bounds are not displayed.
𝛿+
𝛿−
Ignore data
points outside
the bounds
38

39. 39

40. Conclusions
 The physical constraints on the Thomsen parameter δ can help us understand the relation
between δ and the other Thomsen parameters.
 Generally, δ increases with ε and decreases with increasing γ . Variation of β0/α0 of the
hydrocarbon source rocks in a certain area is usually small so that δ is less sensitive to β0/α0.
 δ can be approximately predicted by the other Thomsen parameters .
 Using these constraints, there exist significant uncertainties in laboratory velocity anisotropy
measurement.