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Wicksell's or Krumbein's corrections based on probability of sectioning spherical grains at random can be used to yield loose-grain size moments from the observed thin-section size moments. Mathematical theory and experimental results clearly demonstrate that the probability of slicing spherical grains is directly proportional to their diameters (Wicksell's assumption) and not equal for all sizes as assumed by Krumbein. Therefore, correlating thin-section and loose-grain mean sizes (made dimensionless by dividing by 1 mm) by Wicksell's procedure and linearizing the equation by applying phi-transformation (^phgr=log2) to both sides, one obtains
where subscripts n and w represent number and weight (volume) frequency, respectively; where R.B. is the residual bias
hm is the harmonic mean; bar above letter indicates arithmetic mean; capital and small letters refer to loose-grain and thin-section sizes, respectively; Roman and Greek letters refer to sample and population values, respectively; P, p are projection (equal projection area and nominal sectional) diameters; A, a are long (circumscribing circle) diameters; B, b are short (inscribed circle) diameters (B is actually loose-grain intermediate diameter); ^Dgr, D, ^dgr, d are spherical diameters; ^phgr(c)n and ^phgr(c)w are Wicksell's correction constants having phi-values of 0.651 and 0.179, respectively. Nine multivariate linear correlation equations can, in general, be established between ^phgr(Phm), ^phgr(a), ^phgr(b), and ^phgr(P), ^phgr(A), ^phgr(B); where ^phgr(c) SUB>n(or w) is a constant. The correlation equation between ^phgr(a) and ^phgr(A), for example, is: ^phgr(a)n(or w) = ^phgr(c)n(or w) + ^phgr(A)n(or w) + ^phgr(R.B.)n(or w) + ^phgr(B/A)n(or w) - ^phgr(phm/a)n(or w).
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