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On Backus average for oblique incidence
David R. Dalton∗
, Michael A. Slawinski †
January 12, 2016
Abstract
We postulate that validity of the Backus (1962) average, whose weights are layer thicknesses, is
limited to waves whose incidence is nearly vertical. The accuracy of this average decreases with
the increase of the source-receiver offset. However, if the weighting is adjusted by the distance
travelled by a signal in each layer, such a modified average results in accurate predictions of trav-
eltimes through these layers.
1 Introduction
Hookean solids, which are commonly used in seismology as mathematical analogies of physical
materials, are defined by their mechanical property relating linearly the stress tensor, σ, and the
strain tensor, ε,
σij =
3
X
k=1
3
X
`=1
cijk` εk` , i, j = 1,2,3,
where cis the elasticity tensor. The Backus (1962) average allows us to quantify the response of a
wave propagating through a series of parallel Hookean layers whose thicknesses are much smaller
than the wavelength.
According to Backus (1962), the average of f(x3)of “width” `0is
f(x3) :=
∞
Z
−∞
w(ζ−x3)f(ζ) dζ , (1)
where w(x3)is the weight function with the following properties:
w(x3)>0, w(±∞) = 0 ,
∞
Z
−∞
w(x3) dx3= 1 ,
∞
Z
−∞
x3w(x3) dx3= 0 ,
∞
Z
−∞
x2
3w(x3) dx3= (`0)2.
∗Department of Earth Sciences, Memorial University of Newfoundland, dalton.nfld@gmail.com
†Department of Earth Sciences, Memorial University of Newfoundland, mslawins@mac.com
1
arXiv:1601.02966v1 [physics.geo-ph] 12 Jan 2016
These properties define w(x3)as a probability-density function with mean 0and standard devia-
tion `0, explaining the use of the term “width” for `0.
The long-wavelength homogeneous media equivalent to a stack of isotropic or transversely
isotropic layers with thicknesses much less than the signal wavelength are shown by Backus (1962)
to be transversely isotropic. The Backus (1962) formulation is reviewed by Slawinski (2016) and
Bos et al. (2016), where formulations for generally anisotropic, monoclinic, and orthotropic thin
layers are also derived. Bos et al. (2016) examine assumptions and approximations underlying the
Backus (1962) formulation, which is derived by expressing rapidly varying stresses and strains in
terms of products of algebraic combinations of rapidly varying elasticity parameters with slowly
varying stresses and strains. The only mathematical approximation in the formulation is that the
average of a product of a rapidly varying function and a slowly varying function is approximately
equal to the product of the averages of the two functions.
2 Formulation
Let us consider a stack of ten isotropic horizontal layers, each with a thickness of 100 meters
(Brisco, 2014). Their elasticity parameters are listed in Appendix A, Table 1.
For vertical incidence, the Fermat traveltime through these layers is 229.47 ms. If we perform
the standard Backus average—weighted by layer thickness of these ten layers, as in equation (2),
below—then, the equivalent density-scaled elasticity parameters are hc1111i= 18.84 ,hc1212i=
3.99 ,hc1133i= 10.96 ,hc2323 i= 3.38 and hc3333i= 18.43 ; their units are 106m−2s−2. With these
parameters and for the vertical incidence, the resulting P-wave traveltime through the equivalent
transversely isotropic medium is 232.92 ms, which—in comparison to the Fermat traveltime—is
high by 3.45 ms.
To examine the layer-thickness weighting, let us consider one of the equivalent-medium pa-
rameters,
cTI
1212 =
n
P
i=1
hic2323i
n
P
j=1
hj
,(2)
where hiis the thickness of the ith layer, which herein is 100 m for each layer; thus, each layer is
weighted equally by 0.1.
If we consider a P-wave signal whose takeoff angle, with respect to the vertical, is π/6, this
signal reaches—in accordance with Snell’s law—the bottom of the stack at a horizontal distance
of 1072.89 m. Its Fermat traveltime is 330.58 ms.
If we perform the standard Backus average, the traveltime in the equivalent medium, which
corresponds to the ray angle of 47.01◦, is 343.87 ms, which is higher by 13.3ms than its Fermat
counterpart. If, however, we weight the average by the distance travelled in each layer, as in
equation (3), below, the equivalent elasticity parameters become hc1111i= 20.126 ,hc1212 i=
4.100 ,hc1133i= 12.059 ,hc2323 i= 3.450 and hc3333i= 19.762 . In such a case, the traveltime
is 332.44 ms—which is higher by only 1.9ms—and is an order of magnitude more accurate than
2
using the standard approach. The distances travelled in each layer and the resulting weights are
given in Appendix A, Table 2. In such a case, expression (2) becomes
cTI
1212 =
n
P
i=1
dic2323i
n
P
j=1
dj
,(3)
where diis the distance travelled in the ith layer, which—for vertical incidence—is equal to hi.
According to Lemma 2 of Bos et al. (2016), the stability conditions are preserved by the Backus
average. In other words, if the individual layers satisfy these conditions, so does their equivalent
medium. This remains true for the modified Backus average.
3 Discussion
The Backus (1962) average with weighting by the thickness of layer assumes vertical or near-
vertical incidence. Consequently, such an average does not result in accurate traveltimes for the
far-offset or, in particular, cross-well data, which nowadays are common seismic experiments, and
were not half-a-century ago, when the Backus (1962) average was formulated.
If we modify the weighting to be by the distance travelled in each layer, then the resulting trav-
eltimes are significantly more accurate. Such weighting, however, entails further considerations.
Since the distance travelled in each layer is a function of Snell’s law, there is a need to modify the
weights with the source-receiver offset. However, given information about layers, it is achievable
algorithmically by accounting for distance travelled in each layer as a function of offset.
There is also an interesting issue to consider. The modified equivalent medium is defined by its
elasticity parameters, which are functions of the obliqueness of rays within each layer. This means
that the equivalent-medium parameters are different for the qP waves, for the qSV waves and for
the SH waves. However, since a Hookean solid exists in the mathematical realm, not the physical
world, such a consideration is not paradoxical. It is common to invoke even different constitutive
equations for the same physical material depending on empirical considerations. Furthermore, it
might be possible to derive elasticity parameters of a single Hookean solid—possibly of a material
symmetry lower than transverse isotropy—whose behaviour accounts for both near and far offsets
in the case of three waves.
It is interesting to note that—in each examined case—the traveltime in the equivalent medium
is greater than its Fermat counterpart through the sequence of layers. It might be a consequence
of optimization, which—in the case of layers—benefits from a model with a larger number of
parameters.
There remains a fundamental question: Is the Fermat traveltime an appropriate criterion to
consider the accuracy of the Backus average? An objection to such a criterion is provided by the
following Gedankenexperiment. Consider a stack of thin layers, where—in one of these layers—
waves propagate much faster than in all others. In accordance with Fermat’s principle, distance
travelled by a signal within this layer is much larger than in any other layer, which might be
expressed by the ratio of a distance travelled in a given layer divided by its thickness. This effect
3
is not accommodated by the standard Backus average, since this effect is offset-dependent and
the average is not, but it is accommodated by the modified average discussed herein. However, a
property of such a single layer might be negligible on long-wavelength signal. To address such
issues, it might be necessary to consider a full-waveform forward model, and even a laboratory
experimental set-up.
As an aside, let us recognize that—if we keep the Fermat traveltime as a criterion—making the
propagation speed a function of the wavelength would not accommodate the traveltime discrepancy
due to offset.
Be that as it may, it must be recognized that the discrepancy between the traveltimes in the
layered and equivalent media increases with the source-receiver offset. In the limit—for a wave
propagating horizontally through a stack of horizontal layers—the Backus average, even in its
modified form, is not valid, due to its underlying assumption of a load on the top and bottom only.
Acknowledgments
We wish to acknowledge discussions with Theodore Stanoev. This research was performed in
the context of The Geomechanics Project supported by Husky Energy. Also, this research was
partially supported by the Natural Sciences and Engineering Research Council of Canada, grant
238416-2013.
References
Backus, G.E., Long-wave elastic anisotropy produced by horizontal layering, J. Geophys. Res.,67,
11, 4427–4440, 1962.
Brisco, C., Anisotropy vs. inhomogeneity: Algorithm formulation, coding and modelling, Honours
Thesis, Memorial University, 2014.
Bos, L, D.R. Dalton, M.A. Slawinski and T. Stanoev, On Backus average for generally anisotropic
layers, arXiv, 2016.
Slawinski, M.A. Wavefronts and rays in seismology: Answers to unasked questions, World Scien-
tific, 2016.
Slawinski, M.A., Waves and rays in elastic continua, World Scientific, 2015.
4
Appendix A
layer c1111 vP
1 10.56 3.25
2 20.52 4.53
3 31.14 5.58
4 14.82 3.85
5 32.15 5.67
6 16.00 4.00
7 16.40 4.05
8 18.06 4.25
9 31.47 5.61
10 17.31 4.16
Table 1: Density-scaled elasticity parameters, whose units are 106m−2s−2, for a stack of isotropic layers,
and the corresponding P-wave speeds in km s−1.
layer diwi
1 115.47 0.0773
2 139.45 0.0934
3 195.07 0.1306
4 124.12 0.0831
5 204.61 0.1370
6 126.88 0.0849
7 127.85 0.0855
8 132.17 0.0885
9 198.04 0.1326
10 130.17 0.0871
Table 2: Distances, di, in meters, travelled by the Pwave in each layer, and the corresponding
averaging weights, wi=di/(P10
j=1 dj).
5
