© 2013, PARADIGM. ALL RIGHTS RESERVED. Long Offset Moveout Approximation in Layered Elastic...

© 2013, PARADIGM. ALL RIGHTS RESERVED.

Long Offset Moveout Approximation in Layered Elastic Orthorhombic Media

Zvi Koren and Igor Ravve


Locally 1D Orthorhombic Layered Model

Multi-layer orthorhombic structure Different azimuthal orientation at each layer Common vertical axis Wave type: Compressional

In 1D model, the magnitude and the azimuth of horizontal slowness are the same for all layers

2


Layer Parameters

Thickness and Vertical compressional velocity:

Vertical shear-to-compression velocity ratio:

Thomsen-like parameters:

Azimuthal orientation:

3

ii vz ,P,

iiiiiii ,2,1,2,1,3,2,1 ,,,,,,

2,P

2,S /1

1 iixi vvf

ix ,1


Effective Model

Effective model is presented by a single layer with azimuthal anisotropy that• yields the same moveout as the original

layered medium, for any magnitude and azimuth of the horizontal slowness

4


Direct & Parametric Presentation of Azimuthally Dependent Normal Moveout

Direct NMO vs. surface azimuth/offset: Direct NMO vs. phase velocity azimuth/offset: NMO vs. phase velocity azimuth/horizontal slowness:

5

h

hT

hLpt

ph

ph,,

,

,phs

phs

phs

ht

ht

,

,

phs

off

Lengthwise – offset along phase azimuthTransverse – offset normal to Lengthwise 90phs

phs


Classical Moveout Approximation for ORT

Approximation for VTI, Tsvankin & Thomsen (1994)

Choices for effective horizontal velocity:

Alkhalifah and Tsvankin, 1995:

6

2

22

h

2h4

eff42

42

44

4

22o

22

2o

22

44

2o

22

2

2o

2o

2

,24 VV

VAA

V

VVA

hAtVtV

hA

tV

h

t

tt

iii VVVVVV ,hh,hh,hh RMQ,RMS,MAX

eff21 A

works for azimuthal anisotropy as well


Why Parametric Approximation?

Fractional moveout approximation has the asymptotic correction factor in the denominator of the nonhyperbolic term, A

While it works perfectly for compressional waves in VTI layered medium, it may lead to negative correction factor A for ORT medium

It never happens in ORT planes of symmetry, but may occur for some azimuths in between

7


Effective Model for Short Offsets

Short-offset effective model has 8 coefficients:• 3 low-order coefficients: fast and slow NMO

velocities and slow azimuth• 5 high order coefficients: three effective

anellipticities and two additional azimuths

8


Short-Offset Model for Fixed Phase Azimuth

Generally (for any azimuth), there are eight short-offset moveout coefficients

For a fixed phase velocity azimuth, there are only one low-order and one high-order short-offset coefficients

9


Effective Model for Fixed Azimuth

The short-offset coefficients are related to power series expansion for infinitesimal horizontal slowness

To accurately describe the moveout, expansion coefficients are needed also in the proximity of the critical slowness (long-offset parameters)

10


Long-Offset Parameters: Per Azimuth

Short-offset parameters are computed in the proximity of the vertical direction, it has no azimuth

Long-offset parameters are computed for a proximity of a horizontal direction, characterized by a fixed azimuth. We compute them per azimuth

11


Separation of Long-Offset Parameters Two long-offset coefficients enforce convergence of

moveout to correct asymptote for infinite offsets Unlike short-offset parameters related to all layers, the

two long-offset parameters are separated The first is related to the “fast” layer (with fastest

horizontal velocity for given phase azimuth) and controls the tilt of the asymptote

The second is related to all other (“slow”) layers and controls the elevation of the asymptote

12


Gluing (Combining) the Coefficients

With the given short-offset and long-offset coefficients, we can “glue” them into a unique continuous function for the whole feasible range of the horizontal slowness

Expansions of the synthetized function into a power series for the infinitesimal horizontal slowness and in the proximity of the critical slowness yield the required computed coefficients

13


Short-Offset Expansions

Expansions of three moveout components for infinitesimal horizontal slowness

14

crit

442o

53

53

)Transverse(

)Lengthwise(

p

pp

pOpbpatt

pOpbpah

pOpbpah

hh

hhtht

hhThTT

hhLhLL


Short-Offset Coefficients

15

4critphs44phs44phs42phs424phs

2critphs2phs22phs


critphs2phs2phs


critphs2phs22phs

4sin4cos2sin2cos4

3

,2sin2cos2

1

Traveltime

4cos4sin2cos2

12sin

2

1

,2cos2sin

componentoffset Transverse

4sin4cos2sin2cos

,2sin2cos

componentoffset Lengthwise

pWWWWUb

pWWUa

pWWWWb

pWWa

pWWWWUb

pWWUa

yxyxt

yxt

yxyxT

yxT

yxyxL

yxL


Long-Offset Expansions

Expansions of three moveout components in the proximity of critical slowness

Unbounded term includes the small value in the denominator

21

t

hh

h

t

TT

T

LL

L

p

pp

p

Ottc

t

Ohzc

h

Ohzc

h

cri

2

slowfasto,

slow,fast

slow,fast

1,1


How We Compute Contribution of Slow Layers

Contribution of “slow” layers in the lengthwise and transverse offset components and traveltime is computed per slowness azimuth:• Assume in the “fast” layer propagation occurs in the

horizontal plane, zenith angle 90 deg.• Applying Snell’s law, compute zenith angle of the

phase velocity for each “slow” layers• Given phase velocity direction, compute for each

“slow” layer

22

slowslow,slow, ,, thh TL


Contribution of Fast Layer into Moveout

We assume that the phase velocity direction in the local orthorhombic frame of the “fast” layer is , where the vertical component is infinitesimal, and the horizontal components are

is the local ORT axis of the “fast” layer

23

32,1 ,nnn

23phs2

23phs1 1sin,1cos

11nnnn xx

1x


Contribution of Fast Layer (Continued)

The horizontal slowness in the “fast” layer is

Performing the infinitesimal analysis, we obtain the phase velocity, the polarization vector, the ray velocity components, the components of the lateral propagation and the traveltime vs. the infinitesimal parameter

24

321phs

23

phsphsphs

phs2

phsphsphs

phs

,,

1

,

cos1

,

sin

nnnV

n

VVph

2crit

2 /1 pph


From Coefficients at Two Ends of Slowness Interval to Combined Continuous Moveout

The offset components and the traveltime are approximated for the whole feasible range of horizontal slowness with continuous functions

Coefficients of continuous functions are obtained by combining short- and long-offset coefficients

25


Combined Moveout Functions

The moveout approximation functions are

26

2

crit

4242o

33

33

1

10

h

h

hh

hthththt

hThThThTT

hLhLhLhLL

p

pp

pp

pDpCpBpAtt

pDpCpBpAh

pDpCpBpAh


Intercept Time

With the parametric functions, we obtain the moveout approximation in domain in a straightforward way

The horizontal slowness has no transverse component while the offset has both components

The intercept time simplifies to

28

hhhhh ppptp hp

p

hLhhh phpptp


Test for Multi-Layer Structure

29

# δ1 δ2 δ3 ε1 ε2 γ1 γ2 V, km/s f Δz, km φaxo

1 0.15 0.15 0 0.20 0.20 0.06 0.06 2.0 0.72 0.5 VTI

2 0.12 -0.08 0.07 0.18 -0.15 0.03 -0.03 3.0 0.74 0.5 20

3 0.09 -0.10 -0.06 0.16 -0.12 0.10 -0.08 4.8 0.75 0.5 110

4 0.13 -0.08 -0.07 0.15 -0.14 0.08 -0.09 3.0 0.76 0.5 60

5 0.16 -0.17 0.05 0.12 -0.17 0.05 -0.06 3.5 0.78 0.5 140


Lengthwise and Transverse Offset Components vs. Slowness for Constant Phase Azimuth

30

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.2 0.4 0.6 0.8 1.0

Nor

mal

ized

off

set c

ompo

nent

s

Normalized horizontal slowness

Lengthwise & transverse offset componentsvs. horizontal slowness, ψphs=0

Exact hxExact hyApprox hxApprox hy

7.6θphs , deg

15.3 23.2 31.3 40.00

-8.E-03

-6.E-03

-4.E-03

-2.E-03

0.E+00

2.E-03

0.0 0.2 0.4 0.6 0.8 1.0

Rel

ativ

e er

ror


Relative error of lengthwise & transverse offset components vs. horizontal slowness, ψphs=0

Error hxError hy

7.6θphs , deg

15.3 23.2 31.3 40.00


Lag between Acquisition Azimuthand Phase Velocity Azimuth

31Const. reflection angle, varying azimuthConst. phase azm, varying reflection angle

2

4

6

8

10

12

0.0 0.2 0.4 0.6 0.8 1.0

Azi

mut

hal l

ag, d

eg


Lag between acquisition and phasevelocity azimuths, ψphs=0

ExactApprox

7.6 θphs , deg.15.3 23.2 31.3 40.00

-6

-4

-2

0

2

4

6

8

10

0 30 60 90 120 150 180

Azi

mut

hal l

ag, d

eg

Phase velocity azimuth, deg.

Lag between acquisition and phase velocityazimuths vs. phase velocity azimuth, θphs=38o

ExactApprox


Traveltime vs. Slowness & its Error for Constant Phase Velocity Azimuth

32

traveltime vs. slowness error of traveltime vs. slowness

1.00

1.25

1.50

1.75

2.00

2.25

2.50

0.0 0.2 0.4 0.6 0.8 1.0

Nor

mal

ized

tim

e


Traveltime vs. slowness, ψphs=0

ExactApprox

7.6θphs , deg.

15.3 23.2 31.3 40.00

0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

1.0E-02

1.2E-02

1.4E-02

0.0 0.2 0.4 0.6 0.8 1.0

Rel

ativ

e ti

me

erro

r


Relative error of traveltime vs. slowness, ψphs=0

Error

7.6θphs , deg.

15.3 23.2 31.3 40.00


Traveltime vs. Offset for Constant Phase Azimuth: Parametric Model and Alkhalifah Strong & Weak

33

traveltime vs. offset error of traveltime vs. offset

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 1.0 2.0 3.0 4.0

Nor

mal

ized

trav

elti

me

Normalized offset

Traveltime vs. offset, ψphs=0

ExactA StrongA WeakApprox

θphs , deg.

36.6 38.80 39.5 39.7

-0.01

0.00

0.01

0.02

0.03

0.04

0 1 2 3 4

Rel

ativ

e er

ror

of tr

avel

tim

e

Normalized offset

Relative error of traveltime vs. offset, ψphs=0

A StrongA WeakApprox

36.6

θphs , deg.

38.8 39.50 39.7


Lengthwise and Transverse Offset Components vs. Phase Azimuth for Constant Reflection Angle

34

lengthwise & transverse offset components error of offset components

-0.5

0.0

0.5

1.0

1.5

0 30 60 90 120 150 180

Nor

mal

ized

off

set c

ompo

nent

s


Lengthwise & transverse offset componentsvs. phase velocity azimuth, θphs=38o

Exact hxExact hyApprox hxApprox hy

-1.6E-02

-1.2E-02

-8.0E-03

-4.0E-03

0.0E+00

4.0E-03

0 30 60 90 120 150 180

Abs

olur

e er

ror

of o

ffse

t com

pone

nts


Absolute error of lengthwise & transverse offset components vs. phase velocity azimuth, θphs=38o

Error hxError hy


Traveltime vs. Phase Azimuth for Constant Refl. Angle: Parametric and Alkhalifah Strong & Weak

35

traveltime vs. phase velocity azimuth error of traveltime vs. phase azimuth

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

0 30 60 90 120 150 180

Nor

mal

ized

tim

e


Traveltime vs. phase velocity azimuth, θphs=38o

ExactA StrongA WeakApprox

0.00

0.01

0.02

0.03

0.04

0 30 60 90 120 150 180

Rel

ativ

e tr

avel

tim

e er

ror


Error of traveltime vs. phase velocity azimuth, θphs=38o

A StrongA WeakApprox


Conclusions

We derived new asymptotic correction of the moveout approximation for ORT layered media

The approximation has the same power series expansion of the moveout components as the moveout of the original multi-layer package for• infinitesimal slowness and• nearly critical slowness

40


Conclusions (continued)

For infinitesimal slowness, we keep two terms of the moveout series per azimuth (and vertical time)

For nearly critical slowness, we keep two terms of the moveout series per azimuth as well

One long-offset term characterizes the propagation through the layer with the fastest horizontal velocity, while the other term describes the propagation through the “slower” layers

41


Conclusions (continued)

The approximation is parametric: lengthwise and transverse offset components and traveltime are functions of horizontal slowness and its azimuth

Parametric functions allow approximating the moveout in both t-x and tau-p domains

For wide opening angles the asymptotic correction terms are essential to match the exact ray tracing

42

© 2013, PARADIGM. ALL RIGHTS RESERVED. Long Offset Moveout Approximation in Layered Elastic...

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