Petroglyphs at Cochineros on the banks of the river Mala in central Peru have been pecked or pounded into a glossy black rock varnish. There are distinct differences in brightness between different elements, readily visible to the naked eye.
Digital photography combined with image analysis allows the brightness to be quantified as average pixel intensity of an element. The absolute values of the measurements vary from one photograph to another, but the hierarchy or ordering of the images from most to least bright is robust.
I assume that two processes are causing the change in the image.
The first is that the bright white surface crystalline surface fades over time due to weathering. I model this as an exponential decay so that there is a probability alpha that a white pixel will become grey, the colour of the unmarked rock.
If there areNo white pixels at time t=0, then a certain fraction will have become grey, the colour of the unmarked rock given by
N = Noexp(-alpha(t))
Where the decay constant alpha is to be determined.
At the same time, there is a different probability that rock varnish grows on a pixel, beta, and the fraction that then becomes black.
N = Noexp(-beta(t))
Close up analysis of the rock surface and measurement of the using the opensource Fiji image processing and analysis software.
Three areas of a digital photograph were analysed. The first, bright white, is a modern graffiti with the date 1964 (for reasons explained elsewhere I assign this the date 1980 or 40 years BP.
The second is a spiral on top if which the date was drawn, and the third is the unmarked black rock which is smooth and glossy.
Histograms of pixel intensity were plotted using the fiji software (Figure ) and then saved as spreadsheets of the number of pixels with each pixel value from 0 to 255.
These were then counted to assign a white, grey and black percentage to the date, the spiral and the black rock. White was taken to be any pixel value from 239 (the maximum value) to 166, grey from 165 to 93, and black from 92 to 18 (the minimum)
The results were as follows
| Bare rock | Spiral | Date | |
| white 238-169 | 1 % | 14 % | 50 % |
| grey 168-100 | 28 % | 48 % | 44 % |
| black 99-18 | 71 % | 38 % | 6 % |
| Bare rock | Spiral | Date | |
| white 238-166 | 1.5 % | 15% | 53% |
| grey 165-93 | 37% | 55% | 43% |
| black 92-18 | 61% | 32% | 4% |
For a material that has two decay routes, the number of remaining nuclei after a time t is
- N = Noexp(-(alpha+ beta)(t))
Where alpha and beta are the two constant above for the probability of a pixel turning grey (through weathering) or black (through growth of rock varnish)
This is a standard equation used for potassium-argon dating where there are two possible decay modes, to Argon or Calcium.
Similarly the number of black pixels after time t is
2. Nb = (beta/(alpha + beta)) No(1 – exp(-(alpha + beta)t))
And the number of grey pixels after time t is
3. Ng = (alpha/(alpha + beta)) No(1 – exp(-(alpha + beta)t)).
Using the data in the above tables, where the percentages represent N0 (white) Ng (grey) and Nb (black) we can obtain estimates for alpha and beta.
I am assigning an age for the spiral of 1200 years.
From 1 N/No = 15/53 = exp((alpha+ beta)(t))
Giving alpha + beta = 0.0011
From 2 Nb/No = (32-4)/96 = (beta/(alpha + beta)) x (53-15/53)
So = (beta/(alpha + beta)) = 0.41
and beta = 0.00045, alpha = 0.00065
Using these values, I can now create a calibration curve for the measured pixel intensity, assigning values to white, grey and black of 220, 130 and 116 (based on measurements on 4602.jpg).
The curve gives a reasonable fit for the calibration points at 40 years (206 compared to 209 +-10 as measured) and 440 years (176 compared to 192+-10) and for the morph group estimated at 1000 years (151 compared to 155+-10).
More interestingly, if this modelling approach is valid, the age of an element can be calculated from the histogram of pixel intensity of an element without the need for calibrators, by using the values of alpha and beta, in the same way as Potassium-Argon dating can determine the age from the ratio of Argon to Calcium decay products.
Note that the values of alpha and beta are based on direct measurement of the petroglyph wear from close up photography, plus an assumed age for one element, the spiral. That spiral date can be deduced/induced/ from the anthropomorphs around the peak of the rock which similar
It is independent of the dsting using thev calibration points for 40 and 440 years ZBP
The dating is therefore independent of the two calibration points at 40 and 440 years Before Present.
From the above equations it can be shown that age t is given by
t = (1/(alpha + beta))ln (alpha+beta)/beta)(Ng/N +1)
Where N is the number of white pixels visible.
t= 1/0.0011 ln 0.0011/0.00065 ((28/15)+1) = 1400 years.
The agreement is hardly surprising since I have input the numbers derived from assuming an age of 1200.
However, this method can now be used on elements on other panels, which in some cases we have reasonable date estimates from iconography. This assumes that the parameters alpha and beta derived from one panel can be transferred to another, that is that the rate of weathering and rock varnish growth are the same.
eg the Chimu birds,
4437 white, 4167 grey and 49 black pixels
The age is then given by
t = (1/(alpha + beta))ln (alpha+beta)/alpha)(Ng/N +1)
using alpha =0.00065, beta = 0.00045, Ng =4167 and N = 4437
T = 909 ln (1.69 x( (4167/4437)+1)) = 454 BP or 1566 CE.
A repeat using the wand tool to measure a bigger area gave
52,913 white, 30,105 grey and 194 black pixels
Giving
T = 909 ln (1.69 x( (30,105/52913)+1)) = 886 BP or 1134 CE.
for the grey value and
T = 909 ln (2.44x( (194/52913)+1)) = 900 BP or 1120 CE.
The chimu felines
The Ychsma birds