Steve, Heather, Mark, and others interested,

        First I'd like to make a point about inversions using the stress
hypothesis vs the instantaneous strain (or strain rate) hypothesis,
then I'll comment on Steve's questions.

                Both the stress and instantaneous strain tensors are second rank
symmetric tensors, and the components of both tensors plot as Mohr
circles. The mathematics for calculating the orientation of the
direction of maximum resolved shear stress on a plane is identical to
that for calculating the orientation of the maximum resolved
instantaneous shear on a plane. Thus, an inversion process that
minimizes the misfit between observed slip-directions and the
theoretical orientations for either maximum resolved shear stress or
maximum resolved instantaneous shear gives exactly the same answer.
Thus all published 'paleostress' inversions can equally well be
interpreted as 'instantaneous-strain' inversions. The fundamental
question, however, is, which tensor does the solution actually
represent, because in nature, the principal axes of stress and strain
are not necessarily parallel, nor is the ratio of differences for the
two sets of principal values necessarily the same.

        With regard to the paper Steve mentioned by Sibson and Xie:
                They assume plane strain, and they assume a horizontal maximum
compressive stress. Neither condition necessarily represents the
actual situation in the earth, so the histograms should be taken with a
grain of salt. Moreover, might their histograms possibly be
interpreted as a near-normal distribution with a maximum at about 45°,
consistent with maximum resolved shear, but not with the Coulomb
fracture criterion? Admittedly there is actually a 'hole' in the
histogram at 45°, but then their sample size is only 31, so
irregularities in the distribution are not unlikely. I also agree with
Heather's comments. So while the paper is an interesting
investigation, in terms of settling the question at hand, it seems too
limited by the assumptions to be very reliable.

        With regard to Lisle and Srivastava, 2004:
                Lisle and Srivastava have shown some consistency between their
predictions based on friction and the observations of natural
fault-slip data, from which they conclude that the natural fault-slip
data reflect the orientation of the principal stresses. There is,
however, no measure of how good the correspondence is. It seems to me
a better test would have been to take the observed shear-planes, use
the friction model to calculate the theoretical direction of shear on
those planes, and measure the misfit between the theoretical and the
observed slip directions. That would at least have given a
quantitative measure of the correspondence between the theory and
observation. As it is, all we are given is contoured plots of
theoretically chosen shear-planes and their slip-directions to compare
with scatter plots of observed shear-plane/slickenline data. Why not
contour the scatter plots of the data? It is difficult to see from the
scatter plots where the maxima really are; contouring would have made
this more obvious. Is it possible that the correspondence is actually
not as close as it seems?

        Despite these criticisms, it may be that these studies, however
imperfect, are pointing toward the validity of a stress interpretation
of fault-slip data. To that extent, they are interesting papers, but
to me they are not really conclusive. Testing the difference between
the hypotheses is difficult because any possible means of independently
measuring the stress depends on knowing some constitutive relation
between stress and the instantaneous strain or (equivalently) the
strain rate. There are some data that suggest the strain
interpretation is better, but they are not conclusive either. Other
tests should be made, including investigations of the rheology of
cataclastic flow. (I know the modelers would be delighted if we found
that the brittle crust actually behaved as an isotropic linear
newtonian viscous material!) But we don't really know at this point,
and I think it is a significant question for research.

        There is also the question of how good a fit is good enough. Does it
matter all that much if the 'true' stress axes are, for example, 20°
away from where the stress interpretation of the fault-slip inversion
says they are? The confidence limits for the inversions commonly allow
at least that much latitude. Perhaps for the uses these results are
put to (e.g. the world-wide 'stress' map, which is really a strain
map), it does not matter that much, but for me it is a question of good
science, and really understanding the system one is trying to work
with. Wrong hypotheses that give a 'good enough' answer will certainly
not lead to reliable progress in our understanding.

        I don't have any vested interest in whether or not the stress axes are
parallel to the instantaneous strain axes, but I do think we should
know whether they are or not. Until we have a good answer, my bias is
that researchers should either use the strain axes as the most direct
interpretation of the inversion solutions, or at least state up front
that they assume an isotropic linear constitutive relation for
cataclastic flow to interpret the inversion solutions as stress axes.

        Apologies for the length of this reply. For those interested, most of
these points have been discussed at even greater length in Twiss and
Unruh, 1998, Jour. Geophys. Res.

        Cheers,

Rob Twiss


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             Robert J. Twiss, Prof. Emeritus email:
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             Geology Department telephone:
(530) 752-0179
             University of California at Davis FAX:
(530) 752-0951
             One Shields Ave. website:
www.geology.ucdavis.edu/
             Davis, CA 95616-8605, USA
faculty/twiss.html

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