# Moveout correction and parameter estimation for horizontal VTI

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```					Azimuthal -p analysis in anisotropic media

Samik Sil and mrinal k. sen Edger Forum, Feb-27,2007

Talk overview
• VTI and HTI media. • NMO Correction and parameter estimation processes in VTI and HTI media (x-t domain). • Tau-p transform and its importance. • Development of new tau-p equation for HTI medium. • Simulated case studies. • Conclusions.

VTI and HTI media
•Anisotropy due to layering •Most common anisotropy in reservoir •Found in shale formations •Vertical axis of symmetry •Anisotropy due to vertical cracks • Common anisotropy, found in fractured medium. •Horizontal axis of symmetry •Wave velocity varies with azimuth The stiffness matrix of VTI and HTI media have only 5 independent elements.
VTI

HTI

VTI and HTI Media parameters
• Thomsen (1986) and Tsvankin (1998) introduced 5 parameters related to stiffness matrix elements to characterize a VTI and HTI media, respectively:
 0 or  
h 0

c33

Vertical P wave velocity



 0 or  0h 
h

c55



Vertical S wave Velocity Fractional difference between horizontal and vertical P wave velocities Second derivative of P wave velocity at vertical incidence

c c  or   11 33 2c33
h

(c13  c55 ) 2  (c33  c55 ) 2  or   2c33 (c33  c55 )

 or  h 

c66  c55 2c55

Fractional difference between horizontal and vertical S wave velocities

Travel time analysis: NMO and parameter estimation in x-t domain

NMO in anisotropic Media
• (x,t) NMO Equation in Isotropic Media
t 2  x   t 2 0  x2
2 vrms

 O( x 4 )

(1)

• (x,t) NMO Equation in VTI and HTI Media (Thomsen, Tsvankin, Alkhalifah)
2 tT

 A0  A2 x  A4 x  ,
2 4

(2)

1 A0  t , A2  2 ,  0 1  2 
2 0

For VTI

For HTI

where

NMO correction and parameter estimation is done by curve fitting.

Remarks
•Equation for traveltime is approximate and uses rms parameters. •Truncated non hyperbolic terms may be required even for isotropic media to model traveltimes at large offsets.

•When a ray transmits through an overlying anisotropic layer, x-t curve fitting method, results in misinterpretation. HTI
ISO

• These problems can be solved by converting x-t domain data in tau-p domain.

x-t to -p conversion
X


T
event

Slant Stacking
Q( , p)   dt     px  t  P  x, t  dx
 0  

Why is it called ‘slant stacking’?

t    px;

  t  px

- It sums amplitudes along a line of constant slope on (x,t) space

An example of tau-p converted seismogram

(  , p ) NMO Equation in Anisotropic media
i i   p    zi  qup  qdown  NL i 1

In general

qup  qdown qup  qdown

In an isotropic or VTI,HTI medium Therefore
  p   2  zi qi
i 1 NL

i.e. for two layer case, tau-p curve of the 2nd layer will be obtain by summing with the tau values from the 1st layer

This helps to introduce layer striping approach of NMO correction and parameter estimation

…And also the exact equation
• Van der Baan and Kendall (2003) obtain exact equation for tau-p domain both for VTI and HTI medium.

•VTI Equation

•HTI Equation

Problem with exact equation
•many parameters are not constrained by data. •Inherently non-unique, due to large number of parameters. •Difficult to implement.

Solution: Approximate equations with reduced parameters

Previous solutions (Approximate equations)
Example of approximate equation (VTI): Sen and Mukherjee, 2003, Number of reduced parameters=2

Example of approximate equation (HTI): Baan and Kendall, 2003, Number of reduced parameters=4

Results from approximate equations
VTI Medium

HTI Medium
Approximate tau-p equation performed better curve matching (thus NMO correction and parameter estimation) for HTI medium. Baan and Kendall, 2003

Drawback of approximated tau-p equation for HTI medium
• The existing equation has 4 parameters that are not well constrained from single azimuthal measurement. • Fit with exact data is not good enough for performing better NMO correction. • Different equations for different medium is not desirable.

New azimuthal tau-p equation for HTI medium
So we develop the following 2 parameter equation for HTI media:
i 2 p 4 el4 i 1/ 2 i i  ( pr ,  )   0 (1  p 2 el2 )1/ 2 [1  ] 2 i2 1  p  el i 1 nl

Where

 el   0 (1  2 h cos 2  )
h2 h2

( h   h ) cos 4   (1  2 h cos 2  ) 2

• Has only 2 parameters, so easy to implement. • Reduction in uncertainty space. • One equation for all the TI and Isotropic media.
 el   0 (1  2 h cos 2  )
h2 h2

( h   h ) cos 4   (1  2 h cos 2  ) 2

When Isotropic :  h   h  0; When Eliptical :  h     h and  h   ; When VTI :  h   ,  h   ,  h   , and   0;
3 different azimuth data are sufficient to characterize the reservoir.

Simulated Case studies

Simulated Case studies
• We use 3 models:
2.4 km 2.4 km 2.4 km

1 km
1 km

ISO
HTI

ISO
HTI ISO

1 km
1 km 1 km infinite

ISO
HTI VTI ISO Model 3

infinite

ISO
ISO Model 1 Model 2

All the anisotropic parameters used here for standard anisotropy, based on the work of Xu and Tsvankin, 2006 and Sen and Mukherjee, 2003

Seismograms for the model 1

Plots of x-t and tau-p seismograms for zero degree azimuth.

Seismograms for the model 1

Plots of x-t and tau-p seismograms for azimuth=45 deg

Seismograms for the model 1

Plots of x-t and tau-p seismograms for azimuth=90 deg.

•Since the number of parameters are less, grid search can be employed to find the parameters. •Non-uniqueness of the error space has reduced a lot too. •Example of grid searching (model 1, 0 degree azimuth).

Error analysis

NMO correction

NMO corrected result using our equation and isotropic equation for model 1 for azimuth=0 degree.

NMO correction

NMO corrected result using our equation and isotropic equation for model 1 for azimuth=45 degree.

NMO correction

NMO corrected result using our equation and isotropic equation for model 1 for azimuth=90 degree.

Seismograms for model 2

Plots of x-t and tau-p seismograms for azimuth=0 degree. Tau-p method can fit the curve from isotopic layer very well.

Seismograms for model 2

Plots of x-t and tau-p seismograms for azimuth=45 degree. Tau-p method can fit the curve from isotopic layer very well.

Seismograms for model 2

Plots of x-t and tau-p seismograms for azimuth=90 degree. Tau-p method can fit the curve from isotopic layer very well.

Seismograms for model 3

Plots of x-t and tau-p seismograms for azimuth=0 degree. Tau-p method can fit the curve from the VTI layer very well.

Seismograms for model 3

Plots of x-t and tau-p seismograms for azimuth=45 degree. Tau-p method can fit the curve from the VTI layer very well.

Seismograms for model 3

Plots of x-t and tau-p seismograms for azimuth=90 degree. Tau-p method can fit the curve from the VTI layer very well.

NMO correction results

• NMO correction using our equation in tau-p domain.

Conclusions
• The Tau-p transformation is suitable to perform moveout correction for multi anisotropic layer. • We have developed a 2 parameter approximate equation for HTI medium, suitable for VTI and isotropic too. • Our equation performs better than the published equation by reduction of error space for parameter estimation and by performing better NMO correction. • The equation is developed for horizontal layer case only. • The equation is developed considering weak anisotropy, but still can perform satisfactory NMO correction of standard HTI medium. • The mismatch tends to increase for higher values of p and lower value of azimuth angle. • We have developed an interactive code for velocity picking, using our formula, for real life case.

Acknowledgement
• • • • Dr. Mrinal Sen, Advisor Sponsors of the EDGER forum Rishi Dev Bansal Dr. Robert Ferguson

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