distribution of radii for a single grain, exercise

Grain size distribution in-class exercise.

What is the purpose of this exercise? The purpose of this exercise is to teach you how to construct and analyze histograms in Excel, a fundamental, first step in the analysis of a single variable (univariate analysis). In this case we will look at the variable of grain size, of a single grain. You will also learn about some basic univariate statistical descriptors. In addition, the exercise will introduce you to the importance of grain size analysis in geosciences. Finally, this exercise will give you more experience with the idea of approximation and the importance of assumptions in analysis.

Why is the grain size distribution of geologic materials important? Grain size distribution data is widely used in geology and engineering. The grain size of a particle is crucial in determining whether it is transported in a current or when it comes to rest as a current diminishes. This in turn is important for engineering purposes, such as considering what size rip-rap to put in place to armor a channel bank. Grain size distribution is a fundamental determinant of intergrain porosity and permeability. Grain size is important in an array of chemical reactions because smaller grain sizes increase surface area and aggregate reactivity. Grain size distribution also carries information as to origin - consider the difference between plutonic and volcanic rocks. Example abound where grain size is crucial variable in geoscience and other endeavors. Can you think of some additional examples?

Background for exercise: A sphere is an elegant shape, one that has a radius and a distinct middle that is easily found (do you remember how from your high school geometry class?). A perfect sphere has no variation in its radius. Operationally, one typically measures the radius as the distance from the center to the periphery, and the diameter as the length of a line that passes through the center and goes from one side to the other side of the circle. However, an individual sand or cobble grain shows significant departures from a spherical geometry (see the image of a sand grain below), and this creates an object with variance in the diameter one would measure. The next better geometric approximation for a real grain shape might be an ellipsoid. An ellipsoid has 3 primary axes, each of which would be a diameter by our definition. They intersect at the center. There are also 'other' diameters that exist for other lines that pass through the middle. Real grains of course show departures from a perfect ellipsoid form. Lets define our variable of interest as the radii of a single grain; i.e. the lengths of lines that pass from the center to the periphery.
Crucial to the concept of radius is the identification of the grain center. It turns out that finding the center is not a trivial thing. One can take two perpendicular lines and move the intersection about the grain interior until the center point is equidistant from the margins as measured along a straight line. Then this operation can be repeated with two other perpendicular lines multiple times to hone in on the best center approximation. Another approach would simply be to compute the point where the aggregate distance from that point to the grain boundary is a minimum. This point would define the center and hence the radii in a different way. This can be colved by brute computer force. By the end of this course you would be able to set this exercise up in Excel if you wanted to, but one might argue if it is worth the time. Think of a crescent shaped volcanic glass shard from the wall of a vesicle (gas bubble). Does it have a center? Can a grain have multiple centers? How to proceed in that case? However, many grains are close enough to spherical that the concept of grain size as measured by the grain diameter is meaningful and useful. Experience indicates the approximation is useful.

Diagram showing how in cross section a sphere has equal radii, an ellipse has unequal radii, but predictably so, and a real grain has unequal radii, but in a less predictable fashion.

Thin section seen under microscope of medium grained sandstone.

Assumptions: In order to simplify we will consider a cross section of a quartz grain in thin section. We will assume that this cross section is representative of the grain and that it passes through the middle of the grain and that we can identify the center. Can these assumptions be challenged? It turns out that they can. Imagine an egg shaped grain. When it comes to rest statistically it will likely have its long axis horizontal, just like an egg on a counter top. So the direction of the cross section cut is important. Additionally, whether this cut went through the middle or center of the grain is indeterminate since it was basically a random cut. By picking one of the larger grains in the thin section it is more likely that this section passes near the middle. It is not uncommon that assumptions are not perfectly met. It is also important to recognize and analyze one's assumptions. The level of precision needed is an important consideration here. Since we simply want to get some idea of how the diameter of a single grain tends to vary these assumptions are probably closely enough met. If it become important to be very precise about the grain diameter measure then you might want to proceed differently. Can you think of how? There is an interesting trade off. To get an increase in precision of 10% you may have to expend 100% more effort.

Exercise:

Select which of the three grains above you would like to do.
Select your grain center by eye.
Measure carefully the four radii associated with two perpendicular lines you draw through the center.
Bisect each quadrant (by eye) with a diameter, and measure the length of these 4 additional radius radius. Bisect each angle left with a diameter and measure these (should be 16 more). Bisect once again to get 32 more measurements of radii. You should have 64 measurements in all.
For each radius measurement mark a box with an X above the bin interval the value falls within, on the provided graph paper. Note the bins are each .01 mm wide. The scale of the copies of the grain cross sections handed out in class will be so that 10 cm = 1 mm, or 100:1. Most bins will have more than one reading, and so you stack your x boxes as you have more than one radius that falls within a bin.
Which bin interval has the most boxes? This is an estimate of the mode.
What is the difference between your highest and lowest measurement? This is known as your range.
What will be the difference in the shape or form of the histogram for a very spherical grain versus a very elliptical grain?
Can you think of a grain shape that might give you a positively skewed histogram shape?
Please hand in your results.
Remember as always to ask questions.

@Harmon D. Maher Jr., 2006.