In the normal incidence case of Backus averaging, the layer properties of
thickness, , compliance,
, and mass density,
, are replaced by the corresponding properties
of the equivalent homogeneous medium,
,
,and
:
![]() |
(1) | |
(2) | ||
(3) |
that is, the thickness of is the sum thickness
of the layers,
, and the equivalent medium
mechanical
properties are the thickness-weighted averages of those of the layered medium.
However, these layers can also be described in terms of the layer
properties of one-way travel-time,
, and impedance,
, with slowness,
acting as the means for changing the independent variable
between depth and time. It is well known that:
![]() |
(4) | |
(5) |
and from these:
![]() |
(6) | |
(7) |
and thus:
![]() |
||
(8) |
but , so:
![]() |
(9) |
and by similar reasoning the slowness equivalent:
![]() |
(10) |
and, since =
:
![]() |
(11) |