Testing old and new AVO methods

Chuck Ursenbach

CREWES Sponsors Meeting

November 21, 2003

I. Testing pseudo-linear Zoeppritz approximations: P-wave AVO inversion

II. Testing pseudo-linear Zoeppritz approximations: Multicomponent and joint AVO inversion

III. Testing pseudo-linear Zoeppritz approximations: Analytical error expressions

IV. Using the exact Zoeppritz equations in pseudo-linear form: Isolating the effects of input errors

V. Using the exact Zoeppritz equations in pseudo-linear form: Inversion for density

CREWES 2003 Research Reports

Outline

• New Inversion Methods

• Testing with error-free data

• Analytical error expressions

• Testing on input with errors

• Density inversion

I, II

III

V

IV

Aki-Richards Approximation

2

1sin2

cos2

1 22

2PPR

22/)( 21

Depends on /

Snell’s Law: 12 sin/2

/2sin

21 cos2

1cos2

1

Q

2

1sin2

cos2

1 22

2RA

PPR

2

21

2

212

21

21

2

1sinsin2

coscos2

1coscos4

QR LPPP

Pseudo-Linear expression

Pseudo-quadratic expression

21 cos2

1cos2

1

Q

2

212

2

23

213

3

3

221

2

2

22

2

212

2

2

2121

221

coscos)]2/(1[2

sinsin11

21

2

1

coscos)]2/(1[2

sinsin

)]2/(1[

sin4

/2

coscos

coscos)]2/(1[2

sinsin21

sinsin2coscos2

1coscos4

Q

Q

Q

QRPQPP

Accuracy depends on /

Impedance

• IP =

• IS =

IP/IP / + /

IS/IS / + /

P-impedance contrast is predicted accurately

Comparison of IS/IS predictions

Comparison of RPS inversion for IS/IS

A-R P-L P-Q

RPP 8.6 13 2.0

RPS 8.2 3.2 .22

joint 7.2 3.6 .50

Average %-errors

Section Summary• Accurate Zoeppritz approximations can be

cast into an Aki-Richards form for convenient use in AVO

• Errors in predicted contrasts are strongly correlated with /

• Strong cancellation of error for / + /• Strong cancellation of error for / + / in

Pseudo-quadratic method• Pseudo-linear and Pseudo-quadratic

methods give superior values of IS/IS for RPS and joint inversion

Analytical Inversion

• Observation: Inversion of 3 points of noise-free data, ( = 0, 15, 30 ) gives very similar results to densely sampled data

• Conjecture: Inversion should be semi-analytically tractable (with aid of symbolic computation software [Maple])

• Remark: For inversion of PS data only two points should be required ( = 15, 30 )

• Leave /, /, /, / as variables• Assume their value in coefficients is exact• Evaluate necessary functions at : = 0, 15, 30 where sin() = 0, , ½• Carry out inversion using Cramer’s rule• Expand contrast estimates up to cubic

order in exact contrasts, and up to first order in (/ - ½)

Method

2( 3 1) / 4

S-Impedance contrast error

PP PP

3S S

S SAR exact

2

2

2

2

3

2

0 24 0 71 ( )

1 46 3 41 ( )( )

0 988 1 2 ( ) ( )

1 11 2 72 ( ) ( )

0 0081 0 27 ( )( )

0 85 0 42 ( )

0 65 0 20 ( ) ( )

0 5

I I

I I

2

2

3

2

2 0 049 ( )

0 50 0 38 ( )( )

0 086 0 82 ( ) ( )

0 00037 0 00071 ( )

0 20 0 39 ( )

0 25 0 40 ( ) ( ) ( )

PP PP

3S S

S SPQ exact

2

2

3

0 85 0 42 ( )

0 65 0 20 ( ) ( )

0 50 0 38 ( )( )

0 00037 0 71 ( )

I I

I I

P-impedance contrast errorPP PP

P P

P PAR exact

1

4

I I

I I

PP PP 2

P P

P PPL exact

1

4

I I

I I

PP PP

P P

P PPQ exact

, 4m nl

I IO l m n

I I

Section Summary

• Analytical inversion is tractable

• Cubic order formulae give reasonable representation of error

• Potential use in correcting inversion results

• Rigorous illustration of the superiority of P-wave impedance estimates

Sources of AVO error

• Assumptions of the Zoeppritz equations

• Approximations to the Zoeppritz equations

• Limited range of discrete offsets represented

• Errors in input – R (noise, processing), background parameters (velocity model, empirical relations, etc.), angles (velocity model)

2

2121

22111121

2

3

3

22

11221

2

2

2121

21

2121

21

4

1sinsinsinsin)1(

)cos(sinsin)1(coscos)1()]2/(1[

sin

coscos2

11

2

11

2

11)1(

coscos2

11

2

11

2

11)1(

])2/(1[

sinsin

)cos]1[cos]1([2

1

/

coscos

4

11

cos2

11

2

11cos

2

11

2

11

4

1

cos2

11

2

11cos

2

11

2

11

4

11)cos(cos

4

1

PP

PPPP

PP

PP

PPPP

PP

R

RR

R

R

RR

R

Exact Zoeppritz in Pseudo-Linear form

/ = (/)exact + 0.2

Gaussian noise on R: magnitude 0.01

Section Summary

• AVO inversions can be carried out with the pseudo-linear form of the exact Zoeppritz equations

• Provides a means of examining the effect of individual input errors

• Provides a guide to uncertainty propagation

• Provides a guide to assessing the significance of approximation errors

An exact expression quadratic in /

D

DR ExactPP

2

2121

2

2221121

3

3

2211

221

2

2

21

21

21

4

1sinsinsinsin

)]2/[1(

)cos()cos(sinsin

coscos2

12

12

1coscos2

12

12

1

)]2/[1(

sinsincos

21

21cos

21

21

cos2

12

1cos2

12

14

1

D

0ExactPPR D D

Least-squares determination of /

2( / ) ( / ) 0a b c

a, b, c are functions of

, /, , R ()

2 3

2 22 3 2 0i i i i i i i ii i i i i

a b a c b b c c

/ = (/)exact + 0.2

Gaussian noise on R: magnitude 0.01

Section Summary

• The exact Zoeppritz equation can be formulated to allow least-squares extraction of / by solution of a cubic polynomial

• The / errors from this method are distinctly different from those of 3-parameter inversion

• Random input errors seem to be controlled very effectively in this method