To me, this revision is satisfactory and can be now published as is. I
read also with interest the replies of the authors to my concerns in
matter experimental variogram determination and I decided to respond
to some of them here.
1) Experimental semivariogram values near the origin: we now added to
the semivariogram of samples S1-S3 (Figures 12_new, 14_new, 18_new,
below) insets that focus on the short lag distances (subplots d-f
in the Figures 12_new, 14_new, 18_new below). Now, it can be seen
that near the origin a nugget effect is not observed and the
experimental semivariograms are characterized by a parabolic
behavior (red dots), which implies a continuity (due to the
smoothing occurred when averaging over a slice) of the investigated
property and with a continuous slope. We remind that the data
investigated in the semivariograms are an averaged porosity along
incremental slices (whereas the averaging is applied on binarized
voxel value: ‘1’ – pore or ‘0’ – grain) with a single voxel width
along (x-,y-,z-) directions (Figure 11,13,17 in the manuscript).
The parabolic behavior near the origin is fitted well by a Gaussian
model (blue dashed lines). Therefore, calculating the experimental
semivariogram based on slice-by-slice porosity is an appropriate
tool for variogram interpretation, a tool that for the best of our
knowledge was not used in previous studies using CT data.
I believe the authors fail to recognize that they were trying to
establish a geostatistical model for the pore space variability which
ends up violating fundamental assumptions. In the case of the new
12_new pictures, it should be evident that a gaussian variogram with
such a small effective correlation length (0.1 mm) is not adequate to
describe - in geostatistical sense - the sample at the scale they
would like to. Basically the authors say that trespassing a distance
h=0.1 mm, knowing the slice porosity at location x does not correlate
with the slice porosity located at x+h. How can such a model be
meaningful for a sample which is 3x3x4 mm large? Zooming in the
experimental variogram at such small scale to highlight the actual
"s-shaped" form of the experimental variogram near the origin actually
reinforces my argument. With the available sample discretization and
the slice-by-slice evaluation of variogram, the only actually sensible
conclusion is that we are in presence of pure nugget effect with some
nested periodic fluctuations in directions x- and y-, and that there
is a macroscopic stratification in z- that makes impossible to assume
stationarity of order two of the underlying random variable (slice
porosity).
2) We think, therefore, that regularizing is not needed for
semivariogram calculation based on slice-by-slice porosity of CT
data.
Again I strongly disagree. The regularization *by slice* is not suited
to describe variability in this sample. Possibly, defining a
3-dimensional support (a cube!) for the evaluation of porosity,
support which must be large enough to let some continuity in the
computed porosity when ranging across the whole sample, would give
meaningful results and be suitable for variography at the scale of the
sample, provided that it captures some spatial variability which can
be successfully described by an order two stationary RV.
3) The average distance from peak-to-trough in the semivariogram is
about 0.1 mm, with a similar value to the range of correlation that
was calibrated with the Gaussian model for all samples and in all
directions.
This statement again reinforces my opinion: the chosen support for the
description of local porosity is not adequate to represent spatial
variability at the sample scale. This means that the resulting model
is at best useless in geostatistical sense (since it is basically a
pure nugget with some periodic stratifications). Try and apply, e.g.,
kriging with this model: if you are at a distance of around h < 0.2 mm
you can estimate the next slice as being roughly equal to the known
one, and afterwards there is no usable spatial correlation (since the
variogram goes abruptly to the sill, which is equal to the population
variance). Basically, with this model, kriging would give the same
results as a IDW estimation.
Furthermore, the theory of variogram maintains that it is a monotonic
function of distance. If this is not the case, as with the large hole
effects the authors present in their response, it means that something
fundamental such as, i.e., stationarity of order 2 is violated. A hole
effect is a typical example of that. That can be addressed in many
ways, none of which the authors adequately pursued in the paper
revisions which I received.
4) Author response 5
I meant 3D support, not voxel-by-voxel, which is however already a
better approach. The vx-by-vx approach suffers from some of the same
flaws as the "slice-by-slice" approach, since a voxel is clearly way
too small to be locally meaningful. A 3D support is a cube of voxels,
whose sides are to be adjusted in order to make a "continuous"
porosity emerge when the cube moves within the whole sample. The
resulting "cube porosity" could be possibly used as basis for
variography and inform a spatial model at the sample scale.
Nevertheless, with respect to the new figure 21 in the response,
suddenly there is no gaussian / s-shaped behavior of the experimental
variograms near the origin, contradicting their own previous results,
and actually confirming once more my reserves. Did the authors try and
understand why is that? Maybe the apparent experimental gaussian slice
variogram results from an unsuitable choice of support for the
porosity...?
5) Nevertheless, the semivariogram is a useful tool to estimate
distances of correlation and lack of correlation between
measurements. Determining this range of correlation is useful to
estimate the scale of heterogeneity, and is not directly related to
the directional REV. It is more an apparent range, influenced by
cementation, clay minerals and grains. For S3, this apparent range
is shorter compared to the REV determined by the classical
approach. The shorter apparent range from the semivariogram is in
our opinion clearly related to the grain packing, the only
structural phenomenon at the size of the sample domain under
investigation. As a rule of thumb, one would use at least four (4)
grain diameters as REV edge size.
I clearly agree that variography IS indeed a useful tool, BUT it must
be done properly in order to lead to solid conclusions. I disagree
however with the "physical" explanation of why the author's results
came out like they did, in S3 and in the other stone samples. There is
no doubt that the variograms obtained in this way are not usable,
because they are not representative of a spatial variability at the
desired scale, hence not meaningful, irrespective of the degree of
cementation and clay fractions in their samples. Instead of
"discussing away" my technical critiques to an unusable model with a
rather philosophical physical interpretation, the authors should
improve the fundaments of their variographic model itself. As we all
know, "all models are wrong, some of them are useful". |

I have now received a review of the revised version of your ms. You will see that the reviewer has some significant comments on the use of geostatistics tools and the investigation of your results using variograms.

The comments are framed such that you should be able to address them in a straight-forward manner. However, the reviewer also notes that you should undertake a careful analysis if the geostatistical investigations are indeed necessary. As they stand now, the results and corresponding interpretation are questionable. Should you choose to drop the variogram analysis, the reviewer recommends that the new simulations of permeability within the whole of sample S3 be presented and discussed as part of the main text.

I am looking forward to receiving a revised version of your manuscript.