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For the calcium-carbonate dissolution process, many works can be found in literature concerning theoretical and experimental studies of the erosion on limestone rock surfaces. In particular, the recently published study reported in the reference [Szunyogh ] deals with an accurate mathematical model for carbonate-based rocks dissolution, which takes into account many environmental parameters, as the annual rainfall and the slope of the rock surface with respect to the horizontal plane.

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From equation 7 it can be clearly seen that the velocity W is a function of the local slope of the rock surface.

On the basis of the above model it can be stated that the dissolution velocity on the sample under simulation, in each time step is a function of the local slope of the exposed surface.

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Therefore, notwithstanding that most of the original surface is nearly flat, the walls of the engraved moat undergo increased erosion with respect to the standard situation. Once that a suitable value for the annual precipitation has been set, at the end of each simulation time step the local slope can be calculated from the absolute value of the surface profile first derivative along the x axis.

Of course, in order to eliminate the effects of the surface roughness on the derivative function, the surface profile is previously smoothed using a mobile averaging filtering procedure. These data, interpolated within the plots of the Figures 3b and 3c, allow us to calculate the horizontal correction function Cxb ix for the velocity Vb.

The corresponding vertical correction function Czb iz for the dissolution process may be built in a similar way than that for the freeze-thaw cycles but in this case, due to the accumulation of water, the standard value is located at the upper surface while at the bottom of the engraving the erosion velocity is higher.

In particular reference [Buhmann D. For carbonate limestone the value is about 5.

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As far as the mical dissolution rate is concerned, we used the data reported in reference [Szunyogh ] for the involved physical parameters to calculate the equations 8 and 9. However, as shown in the equation 7the most important input parameter needed by the simulation procedure is the average annual rain fall Qa.

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In the lack of accurate scientific data concerning the local evolution of this parameter back to years and coming from geological analyses, the annual rain fall was provisionally deduced from the historical data of up to eleven meteorological measurement stations located around the central massif of the Apuane Alps and at various different altitudes.

The time range spanned by the data was about 80 years. In the Figure 4 the Qa values for the various stations are plotted as a function of the altitude; the dashed line represents the linear regression of the data set.

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The Figure 5 shows in isometric scales the rock surface cross-sections along the x-direction of two sample models drawn every years. The small dots indicate the borders of the trace as detected by the internal algorithm.

As can be seen from this figure, the engraved signs are still observable after long time erosion notwithstanding the progressive widening of the width, the smoothing of the borders and the lowering of the top surface.

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From the results of the performed simulations a general trend in the evolution of the engraved trace profile may be achieved as shown by the plots of Figure 6a and 6b. As can be clearly seen, the depth tends to slowly increase in time with a linear behaviour and the increasing speed K1 was found to be independent on the original depth ho. For every time step year the obtained values for the height h and the width w were used as input data in the equation 12while the trend parameters a, b, c and d were calculated from the averaging of several simulation tests performed using the same ho and wo parameters and the same average annual rain fall Qa.

The output value of the elapsed time was compared with the initial available value.

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The Figure 7 shows the results of this procedure. Here the relative error percentage between the true elapsed time and the back calculated one using the dating algorithm was plotted as a function of time.

This first result is quite encouraging for the rupestrian archaeology which mostly uses cultural and typological considerations for the chronological analyses. However it must be pointed out that the curves of the Figure 7 are affected by a random error only, having zero mean value and caused by the granular nature of the phenomenon and which is responsible of the differences of the real deformation curves from the corresponding average trend behaviours.

The figure used for the dating experiment are drawn in black and labelled. This flat rock, located at an altitude of meter above the sea level, on the top of a sharp cliff overlooking the sea coast, is one of the sites described in the reference [Bagnoli, P.

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The complete map of the engraved rock is shown in the Figure 8. Here there are several figures of billhooks engraved around a small tub and three crosses, two having a Greek shape four arms with the same length and one belonging to the Latin type. The in situ experimental activity, which consisted of the measurements of several engraved profiles on some figures, were carried out by the author with the precious help of some volunteers belonging to the Archaeological and Speleological Group of Pietrasanta GASP and to the UOEI association of Pietrasanta who also provided the security equipments.

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The dating method was applied to a set of engraved figures composed by five billhooks P1-P5 and the three crosses C1-C3 ; in the Figure 8 the chosen artefacts are traced in black lines and marked with the corresponding label. The cross-sections of the engraved figures were captured using a mechanical profiler made of parallel steel needles with a diameter of 1 millimetre and photographed using a high resolution digital camera with a grid paper on the background see Figure 9.

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AutoCad, in order to easily measure both the height and the width with an estimated resolution of less than microns. For each figure under test, up to ten different profiles were taken just for averaging the data.

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The obtained results are shown in the Table I in numerical form and also in the Figure 10 in the graphical one for an easier reading. In the Table I the first columns indicate the elapsed time from the engraving execution obtained as the mean value of the ten different measurements; the second columns contain the standard deviations of the data while in the third columns the corresponding absolute date calculated with respect to the year of the measurement execution From a careful analysis of the obtained results some important details must be particularly noted.

At first and as it was expected, the dates are progressively more recent with the increasing of the average annual rain fall value just because the higher speed of the mical dissolution which, at least at the altitude considered, is largely predominant with respect of the effects of the freeze-thaw cycles mechanism.

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The second property that must be noted is the substantial uniformity of dates within the two separate groups of signs. The chronologies were collected around the a.

The standard deviation for each date is also indicated by the vertical segments. However the dates concentration for the two groups does not necessarily imply that the signs were engraved in the same time because of the uncertainty of the single date values shown by the vertical segment in each point of the Figure 10 and corresponding to about a century. This approximation is practically the amount of the random error caused by the joined effect of the granular stochastic erosive phenomena and of the differences in the shape of the moat in the various cross-sections of the engraving.

This difference, being well higher than the uncertainty due to the random error, seems to be due just to an historical or cultural reason rather than to a numerical one. Beside the uncertainty of the input data required by the mathematical analysis, we believe that the exposed method is very promising for obtaining reliable absolute dates for rock art, at least and at present under the above described conditions which are limestone rocks, horizontal flat surfaces, open air exposure and figures traced in contour.

Note that, at least to our knowledge, the present proposal is the first example of rock art absolute dating using a specific technique for these artefacts.

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Other attempts reported in literature [Bednarick, R. The reported first experimental application of the dating method yielded results which are completely in agreement with our previous attempt of cultural interpretation of the rock art site [Bagnoli, P.

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This procedure typically occurred just in the first centuries of the Middle Age. Beside we have well in mind that the data obtained for the Billhook Step may be still affected by a systematic error due to the uncertainties concerning some input parameters as the average annual rain fall, however we believe that the obtained chronological analysis of the engravings is substantially consistent and useful to demonstrate the validity of the proposed dating method.

This implies that those interpretations of the billhooks locating them on one hand in the modern age XIX century and in the Etruscan or Roman period on the other one, including the ritual connection with the Silvanus god, can be discarded as completely wrong. Further research developments concerning the present dating method proposal may involve suitable validation activities, i.

Other future improvement of the method may include a refinement of the annual rain fall data by investigating the results of geological resears in the Apuane Alps, a modification of the simulation procedure in order to include in any way also the long-time fluctuations of the annual rain fall and the possible use of more sophisticated techniques for measuring the engraved profile such as laser interferometry.

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Citton, G. Sani G. Szunyogh, G. Buhmann, D.

The kinetics of calcite dissolution and precipitation in geologically relevant situations of karst areas.

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Svensson, U. Bednarick, R. The potential of rock patination analysis in Australian archaeology - part 1.

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