Page 46 - Petrology of Sedimentary Rocks

extrapolation. HENCE, THIS IS THE CURVE THAT MUST BE USED FOR ALL
DETERMINATION OF PARAMETERS. The only disadvantage is that it is even less
pictorial than the arithmetic cumulative curve.

4. Frequency Curve. The frequency curve represents in essence a smoothed
histogram in which a continuous bell-shaped curve takes the place of the discontinuous
bar graph. Again, it is chiefly of pictorial value because no statistical parameters can
be read for it. Although strictly pictorial, it gives a much better picture than this
histogram because it is independent of the sieve interval used and is the best method to
use in dissecting mixed populations into their separate normal distributions. Mathemat-
ically, it is the first derivative of the cumulative curve, and is thus obtained by
measuring the slopes of the tangents to the cumulative curve. To construct it, one
plots a cumulative curve with arithmetic (not probability) ordinates. Now one measures
the slopes of tangents to this curve at various grain-size values. For example, if you
want to find the frquency at a diameter of 2.83#, lay a straight-edge tangent to the
curve at the point where the 2.83@ line intersects it. Measure the slope of this tangent
by noting how much the tangent rises over a horizontal distance of 112 phi unit. This
value then is plotted at 2.834 on the frequency curve, whose vertical scale is stated as
“so many percent per half phi unit” (or half Wentworth grade--both being identical). As
many points are plotted as needed, spaced anywhere along the curve (not necessarily at
the analysis points). Be sure to get all points of inflection (steeper places on
cumulative curve) and all “sags” (flatter places); thus you will accurately determine all
the modes and minimums on the frequency curve. The mode may be fixed accurately by
repeated approximation (page 42).

For very accurate work, a cumulative probability curve should be plotted first,
then from this probability curve, data points may be taken to construct a much more
accurate cumulative curve on the arithmetic graph paper.

Statistical Parameters of Grain Size

For evaluation of sets of samples it is probably best to compare the curves
directly by eye as only in that way can the entire character of the sediment curves be
revealed. But this is inconvenient and furthermore not very quantitative: it is often
difficult to decide whether curve A represents a better sorted or finer sample than
curve 3, or how great the differences are. To solve this problem one resorts to various
statistical measures which describe quantitatively certain features of the curves; these
values can then be tabulated and, it is hoped, certain combinations of values may be
indicative of different sedimentary environments.

There are two basic methods of obtaining statistical parameters. The most
commonly used method involves plotting the cumulative curve of the sample and
reading the diameter represented by various cumulative percentages (as, what grain
size value corresponds to the 25% mark of the sediment--meaning 25% of the material
is coarser than that diameter). In this method, much more accurate results are
obtained if one plots the cumulative curve on probability paper, because of the superior
accuracy of extrapolation and interpolation. The second method, called the method of
moments, is far more complicated and probably of no greater value. It is explained at
the end of this section.

The significance of most of the measures here described is shown graphically on
page 51. Refer to it continually.

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