Hybrid inversion, elastic impedance inversion, and prestack waveform inversionSubhashis Mallick, WesternGeco, Houston, TX, USA

Summary

This paper compares the results of hybrid inversion, elastic

impedance inversion, and prestack waveform inversion on a

single data set. The method used for prestack inversion is a

Monte-Carlo-based optimization method called genetic

algorithm (GA). GA performs a full waveform inversion of

normal moveout (NMO) uncorrected prestack seismic data,

giving a detailed elastic earth model. In hybrid inversion,

elastic models from GA inversion and/or well information

are used to construct low-frequency background trends for

the P- and the S-wave impedance values. These background

trends are then used in the poststack inversion of the

amplitude variation with offset (AVO) intercept and pseudo-

S-wave sections to generate the P- and the S-wave

impedance sections. In elastic impedance inversion, elastic

models from GA and/or well information are used to

generate the low-frequency background trends for the elastic

impedance for a range of incidence angles. These

background trends are then used in the poststack inversion

for the same range of angle stacks to generate the elastic

impedance values for those angles. Once obtained, the

elastic model is extracted from a linear fit to the logarithm

of the elastic impedance.

Comparisons of the elastic models from the above inversion

methods demonstrate that the P-wave impedance values,

extracted from each of the inversions, agree with one

another. The S-wave impedance values, obtained from the

prestack and hybrid inversions, agree with each other quite

well. The S-wave impedance obtained from the elastic

impedance inversion, however, is quite sensitive to noise

and matches poorly with the prestack and hybrid inversion

results.

If there is a need to use inversion as a reconnaissance

exploration tool, it is recommended that a hybrid inversion

be used on large data volumes. Once the results from hybrid

inversion are analyzed, prestack inversion may be run over

selected zones of interest for a detailed analysis. Elastic

impedance inversion is extremely sensitive to noise, and the

only model that can be reliably extracted from elastic

impedance is an acoustic P-wave impedance earth model.

Inversion Example

Figure 1 shows a stacked section from the Andaman Sea,

offshore India. The zone of interest in this data set is the

bottom simulating reflector (BSR), typical of deep-water

data, that marks the boundary between gas hydrate and free

gas. The BSR is also associated with a blanking of reflection

above it, as shown in the figure. In this paper, we will study

the inversion results at a single common midpoint (CMP)

location, marked by an arrow in Figure 1.

Figure 2 shows the P-wave impedance, S-wave impedance,

and Poissons ratio, all obtained from the prestack GA

inversion at the CMP location. A description of the prestack

GA inversion can be found in Mallick (1999). Notice that

the BSR at around 3.2 s is marked by a rise in P- and S-

wave impedance. Immediately beneath the BSR, there is a

significant drop in P-wave impedance, while S-wave

impedance does not drop as significantly. This causes

Poissons ratio to drop to about 0.15. Such a sharp drop in

Poissons ratio is indicative of the presence of free gas

underneath the BSR.

Figure 3 shows a comparison of the results of hybrid

inversion, with prestack GA inversion results. The hybrid

inversion methodology has been discussed in detail in

Mallick et al. (2000). Note that only the low-frequency P-

and S-impedance trends were supplied to the poststack

inversions of AVO intercept and pseudo-S-wave data in the

hybrid inversion scheme. Both P- and S-wave impedance

and Poissons ratio obtained from the hybrid inversion

match GA results quite well.

Connolly (1999) introduced the concept of elastic

impedance, and the corresponding inversion methodology

was discussed by Mallick et al. (2000). For the elastic

impedance inversion applied here, elastic impedance values

for an angular range of 5 degrees to 25 degrees in steps of 5

degrees were computed from the prestack GA inverted

model of Figure 2. Figure 4(a) shows these computed

impedance values. Similar to hybrid inversion, only the low-

frequency elastic impedance trends were used in the

poststack inversion of the respective angle stacks. Figure

4(b) shows the results for two such inversions at 5- and 25-

degree angles compared with the original elastic impedance.

Notice that the poststack inversions match the original

elastic impedance model quite well for these two angles.

Inversion results for the other angles were also similar.

Figure 5 shows the results of a least-squares fit to the

logarithm of the elastic impedance values obtained from the

poststack inversion of the angle stacks. Notice that elastic

impedance inversion extracted the P-wave impedance model

quite accurately, but the estimate of the S-wave impedance

is poor. This poor estimate of the S-wave impedance results

in a poor estimation of the Poissons ratio.

Discussion and Conclusion

Prestack waveform inversion, with its detailed wave

equation modeling, is capable of obtaining a good elastic

earth model. Such an elastic model can be used to define the

low-frequency trend in a poststack inversion process.

SEG Int'l Exposition and Annual Meeting * San Antonio, Texas * September 9-14, 2001

Dow

nloa

ded

01/0

8/15

to 5

.22.

98.4

2. R

edist

ribut

ion

subje

ct to

SEG

licen

se or

copy

right;

see T

erms o

f Use

at htt

p://lib

rary.s

eg.or

g/

Hybrid, elastic, and prestack inversion

2

In this paper, two poststack inversion methods are discussed.

Both methods used the same low-frequency background

impedance trend from prestack inversion. In hybrid

inversion, AVO intercept and pseudo-S-wave data are

inverted. In the example shown, hybrid inversion extracted

the original elastic models quite well.

The elastic impedance inversion, where angle stacks for

different incidence angles are inverted, estimated the

original impedance values to a reasonable accuracy. A least-

squares fit to the logarithmic elastic impedance values from

poststack inversion, however, results in a poor estimation of

the S-wave impedance. Further analysis of the elastic

impedance inversion shows that to obtain a good estimation

of the S-wave impedance, elastic impedance values must be

exact. Figure 6 shows the same least-squares fit as Figure 5,

where the input elastic impedance values were the original

values computed from the prestack elastic model. Notice

that in Figure 6, elastic impedance inversion obtained the

original model with good accuracy. In practice, it is not

possible to obtain such exact elastic impedance. Even if the

angular elastic impedance values from poststack inversion

look reasonable in Figure 4b, they will result in a poor

estimation of S-wave impedance. The only reliable model

that elastic impedance inversion can provide is an acoustic

P-wave impedance earth model.

To investigate further why elastic impedance inversion gives

a poor estimate of S-wave impedance, it is necessary to go

into the details of the elastic impedance formulation. Given

an interface with P-wave velocities 1, 2, S-wave velocities

1, 2, and densities 1, 2, the elastic impedance E1 and E2for the two media above and below the interface are written

as (Connolly, 1999)

,)sin41(1sin8

1

tan1

11

222 KKE (1)

and

.)sin41(2sin8

2

tan1

22

222 KKE (2)

The parameter K in equations (1) and (2) above is the square

of the average S-to-P velocity ratio across the interface, and

is the angle of incidence. Once the elastic impedance E1and E2 are defined, the P-wave reflection coefficient for the

incidence angle is given approximately as

.)(12

12

EE

EER

(3)

Because the reflection coefficient as defined in (3) has the

same form as the reflection coefficient at normal incidence,

standard poststack inversion can be applied to the angle

stacks. However, notice that the elastic impedance

formulation has the parameter K that depends upon the

interface. Poststack inversion does not allow such an

interface-dependent parameter, because this implies that a

given layer must have two different elastic impedance

values, depending upon whether the layer is above or below

an interface. To avoid this problem, an interface-

independent K is used. This is done by minimizing the sum

of the difference between the reflection coefficient R for all

interfaces using an exact reflection coefficient formula, and

the reflection coefficient RK for the same interfaces using the

elastic impedance formulation and a constant K value.

Figure 7 shows such a minimization procedure leading to an

error, which is minimum for a K value between 0.2 and

0.25. Running the elastic impedance inversion with different

K values in this range shows that the extracted P-wave

impedance is insensitive to the choice of K. The S-wave

impedance, on the other hand, is quite sensitive to the value

of K.

In addition to the requirement of a constant K, another

problem with the elastic impedance inversion method is the

assumption of a convolutional model at non-normal angles

of incidence. Poststack inversion of angle stacks implies a

convolutional model at different angles. This is not a correct

assumption. To demonstrate this, Figure 8 shows different

synthetic seismograms for the elastic model of Figure 2.

These synthetics were computed using exact wave-equation

modeling and the convolutional assumption of the

reflectivity series with a wavelet. The reflectivity series in

these convolutional seismograms were computed using (a)

the exact reflection coefficient formula, (b) Shueys (Shuey,

1985) three-term approximation, and (c) Shueys two-term

approximation. Note that none of the convolutional

synthetics match the wave-equation synthetics. Elastic

impedance formulation is equivalent to Shueys three-term

approximation, and the synthetic computed with this

approximation is far from the one obtained with wave-

equation modeling. The assumption of a convolutional

model will indeed result in an incorrect model in elastic

impedance inversion.

References:

Connolly, P., 1999, Elastic Impedance: The Leading Edge,

18, 438-452.

Mallick, S., 1999, Some practical aspects of prestack

waveform inversion using a genetic algorithm: an example

from east Texas Woodbine gas sand: Geophysics, 64, 326-

336.

Mallick, S., Huang, X., Lauve, J., and Ahmad, R., 2000,

Hybrid seismic inversion: a reconnaissance tool for

deepwater exploration: The Leading Edge, 19, 1230-1237.

Shuey, R.T., 1985, A simplification of the Zoeppritz

equations: Geophysics, 50, 609-614.

SEG Int'l Exposition and Annual Meeting * San Antonio, Texas * September 9-14, 2001

Dow

nloa

ded

01/0

8/15

to 5

.22.

98.4

2. R

edist

ribut

ion

subje

ct to

SEG

licen

se or

copy

right;

see T

erms o

f Use

at htt

p://lib

rary.s

eg.or

g/

Hybrid, elastic, and prestack inversion

3

Figure 1: Stacked section from the Andaman Sea, offshore

India. A bottom-simulating reflector (BSR) and a blanking

above the BSR characterize the data. The arrow on the top

of the Figure indicates the location for which different

inversion methods described in this paper are run.

2.5

2.9

3.3

0 2000 4000 6000 0.0 0.2 0.4

Figure 2: P- and S-wave impedance and Poissons ratio

model from prestack GA inversion for the location

marked by an arrow in Figure 1.

Figure 3: P- and S-wave impedance and Poissons ratio

from hybrid inversion, compared with GA results. GA P-

impedance and Poissons ratio are in black and hybrid P-

impedance and Poissons ratio are in red. GA and hybrid S-

impedance are in blue and green, respectively.

Figure 4: (a) Elastic impedance, computed from a GA model

between 5 and 25 degrees. (b) Elastic impedance from poststack

inversion at 5 and 25 degrees compared with true elastic

impedance from GA inversion. GA impedance values are in

black and inverted impedance values are in red.

Figure 5: P- and S-wave impedance and Poissons ratio

from elastic impedance inversion compared with GA

inversion results. Colors used are similar to Figure 3.

Figure 6: P- and S-wave impedance and Poissons ratio

from elastic impedance compared with GA inversion. Input

elastic impedance values are true impedance values

computed directly from the GA model. Colors are similar

to Figures 3 and 5.

(a) (b)

5o25o

25o 50

2.5

2.9

3.3

2.5

2.9

3.3

2.5

2.9

3.3

2.5

2.9

3.3

Tim

e (s

)

Tim

e (s

)T

ime

(s)

Tim

e (s

)

Impedance Poissons ratio0 2000 4000 6000 0.0 0.2 0.4 0 2000 4000 0 2000 4000 6000

Impedance

Impedance Poissons ratioImpedance Poissons ratio

0 2000 4000 6000 0.0 0.2 0.40 2000 4000 6000 0.0 0.2 0.4

Impedance Poissons ratio

BSR

BSR

SEG Int'l Exposition and Annual Meeting * San Antonio, Texas * September 9-14, 2001

Dow

nloa

ded

01/0

8/15

to 5

.22.

98.4

2. R

edist

ribut

ion

subje

ct to

SEG

licen

se or

copy

right;

see T

erms o

f Use

at htt

p://lib

rary.s

eg.or

g/

Hybrid, elastic, and prestack inversion

4

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 0.1 0.2 0.3 0.4 0.5 0.6

0.1

0.2

0.3

0.0

Exponent (K)

0 0.1 0.2 0.3 0.4 0.5 0.6

Err

or

()

10o

20o

30o

Figure 7: Computation of the optimum value of the exponent K for elastic impedance inversion.

5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 352.5

2.6

2.7

2.8

2.9

3.0

3.1

3.2

3.3

3.4

3.5

Angle (Degrees)

Tim

e (

s)

Wave Equation Convolutional (Exact) Convolutional (Shuey 3 term) Convolutional (Shuey 2 term)

Figure 8: Synthetic seismic traces, created with the elastic model of Figure 2, using different modeling methods as shown.

SEG Int'l Exposition and Annual Meeting * San Antonio, Texas * September 9-14, 2001

Dow

nloa

ded

01/0

8/15

to 5

.22.

98.4

2. R

edist

ribut

ion

subje

ct to

SEG

licen

se or

copy

right;

see T

erms o

f Use

at htt

p://lib

rary.s

eg.or

g/