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The AAPG/Datapages Combined Publications Database
AAPG Bulletin
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Projection diameter (P, or p), the diameter of a circle having an area equal to that of the grain area, long diameter (A, or a), and short diameter (B, or b) can be measured for loose grains in a stable position in the gravitational field (capital letters) and for grains in the thin section. The stable position of an irregular loose grain in the gravitational field need not be a unique position, but generally the thickness of the loose grain (C) remains vertical. In such a situation <pl>^horbar<above>B</pl> and <pl>^horbar<above>p</pl> are the best estimators of the arithmetic mean nominal diameters B and p, respectively, on a number-frequency basis. However, <pl>^horbar<above>B</pl> and <pl>^horbar<above>p lt;/pl> need not be unbiased estimators of the nominal diameters if some nonspherical grains are present either in the thin section or in the loose grains.
From Krumbein's theory of thin sectioning of spherical grains,
EQUATION (1)
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where <pl>^horbar<above>d</pl>n and <pl>^horbar<above>D</pl>n are the arithmetic mean nominal diameters of thin section and loose grain sizes, respectively, on a number-frequency basis. The correlation equation for <pl>^horbar<above>p</pl> and <pl>^horbar<above>B</pl>, using eq. 1, is
EQUATION (2)
where R.B. is the residual bias which is equal to the terms under the third bracket. Linearizing eq. 2, by taking the phi-transform (-log2) of both sides,
EQUATION (3)
In a similar way one can obtain nine linear correlation equations between ^phgr(<pl>^horbar<above>p</pl>) or ^phgr(<pl>^horbar<above>a</pl>) or ^phgr(<pl>^horbar<above>b</pl>), ^phgr(<pl>^horbar<above>P</pl>) or ^phgr(<pl>^horbar<above>A</pl>) or ^phgr(<pl>^horbar<above>B</pl>). The correlation equation for ^phgr(<pl>^horbar<above>a</pl>) and ^phgr(<pl>^horbar<above>B</pl>) will be
EQUATION (4)
and that between ^phgr(<pl>^horbar<above>a</pl>) and ^phgr(<pl>^horbar<above>A</pl>) will be
EQUATION (5)
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