Recent developments - 3-D Seismic Interpretation - page 216

A3.3 Elastic impedance inversion

The ideas touched on in the last part of chapter 6 have been further developed in recent years. The
concept of elastic impedance has been refined. We recall that this is a layer property ( E ) such that
the reflection coefficient at an interface between the ( n + 1)th and nth layers at incidence angle θ is

given by

R n = ( E n + 1 − E n ) / ( E n + 1 + E n ) .

The concept of extended elastic impedance was introduced by Whitcombe et al. (2002) to remove
some problems with the normalisation of the quantity as originally defined but also to allow the
extension of the concept to incidence angles beyond those physically attainable. To this end, a new
angle χ is introduced such that

tan χ = sin 2

θ.

Then extended elastic impedance is defined to be

EEI ( χ ) = α 0 ρ 0 [( α/α 0 ) p ( β/β 0 ) q ( ρ/ρ 0 ) r ] ,

ρ 0 are reference values
of the respective quantities, e.g. the average value over the interval of interest. Here p , q and r are
given by

α β ρ α 0 , β 0 , and

where is P-wave velocity, is S-wave velocity, is density, and

p = (cos χ + sin χ ) ,
q =− 8 K sin χ,
and r = (cos χ − 4 K sin χ ) ,

where K is a constant equal to the average value of ( β/α ) 2 over the interval of interest.
Extended elastic impedance (EEI) combines P-velocity, S-velocity and density in different ways
as the angle χ changes, and therefore shows varying sensitivity to lithology and fluid content as a
function of χ . If we have a well in which P-velocity, S-velocity and density have all been logged,
then we can calculate the elastic impedance from the above formula for all angles χ from − 90 ◦ to
+ 90 ◦ . We can then compare the EEI curves to a target log in the well, for example a gamma ray log
(if we want to predict sand/shale ratio) or a fluid saturation log (if we want to predict hydrocarbon
presence). In this way we can empirically identify what angle χ gives us the best prediction of the
target log. An example is shown in fig. A3.9 . Here the best visual match to the saturation log is
around χ = 30 ◦ . In a more detailed study, the correlation coefficient between the saturation and
elastic impedance curves was calculated for a range of different χ values, leading to the conclusion
that

33 ◦ gives the best prediction of gas saturation. In practice, the effects of noise in real seismic
trace data have similar effects on elastic impedance to their effects on reflectivity discussed in chapter
5 (section 5.4 ) , and it would be a good idea to look at results for a range of

χ =

values to see what
angle gives the best expression of fluid effects, in a similar way to that shown for reflectivity in
fig. 5.16 .
There is no guarantee that the χ values that we want to use correspond to physically realisable
incidence angles. We recall from chapter 5 that for incidence angles out to 30 ◦ or so the reflection
coefficient at incidence angle θ is given by

χ

R ( θ ) = R 0 + G sin 2 θ,