Testing old and new AVO methods Chuck Ursenbach CREWES Sponsors Meeting November 21, 2003.
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Transcript of Testing old and new AVO methods Chuck Ursenbach CREWES Sponsors Meeting November 21, 2003.
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