testing old and new avo methods chuck ursenbach crewes sponsors meeting november 21, 2003
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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
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