Statistics, the Colossi at Memnon, and Forensic Geochemistry

Earth Science Extras

by Russ Colson

 

The Colossi at Memnon, Egypt, built in memory of Amenhotep III on the west bank of the Nile River about 3200 years ago during the 18^th Egyptian dynasty. southern Colossus on the left, northern Colossus on the right. Image is from the research article Bowman et al. (1984) The Northern Colossus at Memnon: New Slants, Archaeometry v26, 218-229.

 

Introduction to analytical counting statistics

Most analytical techniques involve counting some type of "signal" in order to figure out the concentration of a chemical component. "Signals" include things like photons (either emitted from the sample or transmitted through it) or particles (such as radioactive particles emitted from a sample).   The electron microprobe, an important and versatile analytical tool for study of solid materials like rocks, glass, and ceramics, works by zapping a tiny sample with a beam of electrons.  The high-energy beam causes electrons in the atoms of the sample to get kicked out of their spots, and then, when an electron falls back into the vacated spot in the atom, an x-ray whose wavelength is unique to that particular atom (element) is released.  The number of x-rays depends on how vigorously the sample is zapped by electrons, the concentration of the element in the sample, and how long one counts the x-rays.   The more x-rays that are counted, the better an estimate of chemical concentration becomes. Thus, 'counting statistics' refers to the process of figuring out the statistical uncertainty of an analysis based on the number of 'counts' of a signal, such as x-rays. As the number of counts increases, the uncertainty in an analytical measurement goes down.

 

You can understand this concept by thinking about flipping a coin. We all know that we expect, on average, to get heads 50% of the time. But in practice, we don't always get 50% heads.

Find a coin and flip it 4 times, recording the number of times that it comes up heads. For example, suppose after your four flips you have recorded 3 tails and one heads, yielding 25% heads. Do it again. Perhaps this time you get 2 heads and 2 tails. Repeat 2 more times, recording your results.

Notice how much range there is in the percentage heads. You might get 50% heads about half the time, but the other half of the time you get either 25% or less heads or 75% or more heads.

Now repeat the same excercise, but flip the coin 10 times. Notice that there is smaller chance of getting exactly 50% heads, but there is also a much smaller variation in the percentage that you record. In this excercise, you might expect to get less than 25% or greater than 75% heads only a little more than 10% of the time, much less than with only 4 flips. Try it!

If you flipped a hundred times, the percentage heads would be even more tightly constrained.

 

In analytical counting statistics, the uncertainty in our counts of the 'signal' equals the square root of the number of counts, such that

uncertainty =

The percentage uncertainty equals the uncertainty divided by the number of counts (times 100% to convert to a percentage). or

% uncertainty =

You can see that the uncertainty gets larger as the number of counts increases, but the percentage uncertainty gets smaller.

 

The uncertainty calculated by the formula above is sometimes referred to as 1-sigma uncertainty. For our purposes, we can think of this value as defining a range within which repeated measurements will lie 68.3% of the time. Said another way, the 'true' value will lie within plus or minus 1-sigma of our measurement 68.3% of the time.

If we double this number, 2-sigma, this gives us the range within which repeated measurements will lie 95.4% of the time (or the 'true' value will lie within plus or minus 2-sigma 95.4% of the time). 3-sigma, three times this value, gives the range within whch 99.7% of repeated measurements will lie.

 

In science, the 95% value is considered an important threshold. Any measurement that is not signficant at the 95% confidence interval may not be considered to be significant or meaningful. For example, for an analysis to be signficant, it must indicate a concentration greater than 0. We might say that a measurement is significant at the 95% confidence interval if our measurement is more than 2-sigma away from 0, otherwise, we have not proven that that particular chemical component is even present in the sample. In that case, we would say that the concentration of that element is below the sensitivity of our measurements. We might report the concentration as <x%, where x% is the sensitifity of our measurements, that is, the lowest concentration we can measure where our measured concentration is more than 2-sigma from zero.

 

To calculate a concentration from the number of counts, it is necessary to compare the counts in the sample to counts from a sample with a known concentration of the element that is analyzed in a similar way. The sample with known concentration of the element is called a standard (most chemical analyses involve comparing an unknown sample to standards of known concentration). Ignoring certain complexities, such as how elements other than the element of interest can interact with the x-rays (called matrix effects) and the fact that there are always background x-rays present that aren't directly related to the element of interest, the concentration in the sample/ counts per second = the concentration in the standard/counts per second.

in other words, the rate at which x-rays are generated is directly related to the concentration of the element in the sample if the electron beam and certain other characteristics of the analysis stay the same.

Thus,

concentration in the sample = concentration in the standard x (counts per second in the sample/counts per second in the standard)

 

Because of analytical uncertainties, no chemical analysis is ever exactly correct, but rather yields a range within which the 'true' value probably lies. Which of the following represents the range within which the 'true' value likely lies at the 95% confidence interval? (we are going to ignore the uncertainty in the standard, however, since there are so many more counts for the standard, considering the uncertainty in the standard would not make a very large difference in our result).

 

When considering natural samples, we not only have to consider variability due to analytical uncertainties, but there is often real variation in concentration from one sample to another. Thus, when we consider an 'average' concentration of a set of several samples, the total variation (and accompanying uncertainty) may reflect not only analytical uncertainty, but variability in the samples as well.

 

A Geochemical Story of the Colossi at Memnon

The Colossi at Memnon on the west bank of the Nile River in Egypt were built 3200 years ago in the18^th Egyptian dynasty as a memorial for Amenhotep III, Hundreds of years later, invading armies, hoping to destroy the will of the Egyptians to resist or revolt, tried to destroy these monuments of culture and heritage. Invading Assyrians in the 7th century BC, and Persians in 6^th and 5^th centuries built huge fires around the statues in an effort to cause the stone to crack due to the expansion and contraction resulting from heating and cooling.  Unfortunately for the Assyrians and the Persions, the stone was made of Quartzite. Quartzite is monomineralic--made of only one mineral--and so there was not very much differential expansion from different minerals that would cause cracking. Also quartz (SiO₂) is isometric, meaning its crystalline atomic structure is the same in all directions. Compared to many other minerals, it expands more equally when it gets hot and consequentily does not introduce unbalanced forces that might crack the rock. Also, the coefficient of expansion for quartz is not very high to begin with, meaning it doesn't expand and contract as much as other minerals might have. The Assyrians and Persions failed to destroy the monuments.

Unfortunately for the Egyptians, several hundred years later stil, in 27 BCE, an earthquake succeeded where the Assyrians and Persions failed. The northern of the two Colossi lost the upper half of its body and the back of the throne block.

Two hundred years after the earthquake, according to the Greek Historiian Strabo, the Roman emperor Septimius Severus set about to restore the northern Colossus, which was by that time an ancient work of art from 1400 years earlier.

Did those repairs use Quartzite from same quarry as the original construction, or did the Romans use a different quarry?   Where was the quarry?

Consider the map below (from Bowman et al 1984) showing the location of possible quarries for quartzite. Where do you think the quartzite may have come from, used by either the Epyptians or the Romans 1400 years later? Do you think they brought these immense stones upstream all the way from Cairo, or did they bring them downstream the much shorter distance from Silsila or Aswan?

 

First, let's ask the question of whether the Romans used stones from the same quarry as the Egyptians or whether they used a different quarry. Notice the main 'throne' block in the picture of the northern Colossi (at the top of the page) is composed of a main 'front' piece (this is the remnant built by the Egyptians) and several rear pieces (part of the repair work by the Romans). Are these from the same quarry or not? This is a statistical geochemistry question: are the compositions of stone different from each other at the 95% confidence level, or are they not sufficiently different to determine that they are from different sources? We can conclude that they are different if there is more than 2-sigma between them--or, said another way, if their 1sigma error bars do not overlap (this simplification is strictly only true if the uncertainties for the two values are nearly the same but it is a useful simplification for the puzzles we are going to address).

 

Below is a data table from Bowman et al 1984. Familiarize yourself with the headings on the columns and rows of this table and read the footnotes. Consider what the various numbers mean, and the meaning of the symbol < before some of the numbers (reread the text above if you don't know what this means).

 

Considering additional elements potentially increases confidence in conclusions. Trace elements--that is, elements that are present at low concentrations in the mostly SiO₂ quartziite--can offer additional constraints. Below is another data table from Bowman et al 1984 with measurements of trace element concentrations. Read the footnotes and the headings on the columns and rows of this table. Neutron activation analysis is an important analytical technique for measuring trace element concentrations in solid samples.

 

  

 Let's now return to our starting question: Which quarry was used for the main block (built by Egyptians roughly 3200 years ago) and which was used for the rear blocks (used in repairs by the Romans about 1800 years ago).

Let's first look at the major and minor elements again--we are going to look first at Manganese oxide (Mn). Look over the data table, considering the MnO in particular, and construct an idea of what this one element says about the source for different stones in the northern Colossus. Remember, the uncertainties reported are 1-sigma uncertainties. Write down notes, including your conclusions and your reasons for making those conclusions.

When you have finished the exercise above, test yourself against the question below.

 

 Let's now look at the trace element data. Study the data in the table below and figure out which quarries (Aswan and/or Cairo--Cairo is a subset of Gebel el Ahmar) the main block and rear blocks might be from. Take notes and write down explanatory support for your conclusion being sure to make reference to specific data.

When you have finished the exercise above, test yourself against the questions below.

 

So, perhaps not too amazingly, the original construction brought stone for the Colossi from much farther away and upstream, whereas the Roman repairs invested less effort, bringing stones from closer and bringing them down river.

 

A Fictional Forensic Geochemistry Adventure

Suppose that two manufacturing plants are built in the Karst terrane of southeast Minnesota. These plants both manufacture a sulfur-based pesticide/fungicide (sulfur = S). Remember, this is fictional--such plants are not really there. Being situated on the Karst terrane, any spills or leaks of the pesticide onto the surrounding soil will be washed into the underground cave systems, becoming included in aquifers and also eventually finding their way back out into the surface waters. Because of the convolute plumbing in these cave systems, tracking the source of the pollutant might be quite difficult.

 

When traces of the pesticide are found in caves about 3 miles from the manufacturing plants, and also in streams downflow from the caves, you, as the regional forensic geochemist, are called in to investigate which of the plants, if either, might be the source of this pollution.

 

Although they are manufacturing the same product, the two plants use slightly different manufacturing processes which might introduce subtle differences in the isotopic composition of the S-based pesticide. You decide to begin your investigation by addressing the possibility that the ratio of S32 to S34--two isotopes of sulfur--is different in the product of the two plants. If so, this would give you a fingerprint to distinguish which plant is the source for the pollutant.

As background information, you know that the standard devation of sample (a measurement of 1-sigma uncertainty) can be estimated from the formula

, where = mean value

For the following challenges, you will need to calculate uncertainties using these formulae, or know how to use a spreadsheet to calculate mean and standard deviation (if you don't know how to do this with a spreadsheet, now is a good opportunity to take some time to learn).

 

In addition, you know that if you average several samples (n) that all have the same standard deviation, then the 1-sigma standard deviation of the average value is calculated from the formula

In other words, if you average 4 samples, the uncertainty in the average is 1/2 of the uncertainty of a single sample. If you average 9 samples, then the uncertainty in the average is 1/3 the uncertainty in a single samples, and so forth (note that this is only true if the uncertainties in all of the averaged samples is the same).

 

 

You do three more analyses of the product from each plant and get the data table below. You can either import these data into a spreadsheet to calculate averages and standard deviations, or use the formulae above to answer the following question.

 

 

In order to get more precise ratios for S32/S34 for each of the Plants A and B, you take additional samples from each and analyze them, doing enough samples that the uncertainty becomes very small.

You get the following values (which we will consider to have 'very low' uncertainties due to taking many samples and doing many analyses).

Plant A ratio = 22.59

Plant B ratio = 22.25

 

last updated 6/13//2020. Some data and images from Bowman et al, 1984, The Northern Colossus at Memnon: New Slants, Archaeometry v26, 218-229, as indicated. Other pictures and text are the property of Russ Colson.