which differs only from equation [Cviii] in that p/ rather than /, appears
in the denominator.
 Relationships similar to equations [Ci] to [Cviii] and [Cix] to [Cxv]
for the single-energy gamma-ray attenuation method can also be ob-
tained for the dual-energy method for simultaneous determinations of
soil p and 0. If the attenuated beam intensities for both sources (Ia and
I,) occur with random fluctuations (dla and dIl), then fluctuations will
also occur in determinations of water content (dO) and bulk density
(dp). By differencing equations [Biii] and [Biv] in Appendix B and
adopting criteria similar to that used for Cvii:

 Jsc bLsa
 dO = -/I- v/I [Cxvi]
 X (psa iPwc Psc fPwa)
 /Awc Izwo.
 and dp = /I, v/Ic [Cxvii]
 x (tsga Pwc l-sc Vtwa)
Using standard error propagation theory, equations [Cxvi] and [Cxvii]
become:
 I1/2

 =x c -Ja [Cxviii]
 X (ps) P2wc 2j 12a)
 [(fw2 (i2)2] ^ 2

 and a, = c la [Cxix]
 X (l/sa w -w- 1sc /twa)
where a, and a, represent the minimum resolvable changes in 0 and p
respectively. The relative sensitivities for water content and bulk density
may be calculated, by dividing equation [Cxviii] by [Aiv] in Appendix A
and [Cxix] by [Av] in Appendix A.




 Appendix D. Theoretical consideration of the accuracy and
 precision of water content and density determinations of
 soil by the dual-energy gamma attenuation method

 Theoretical considerations of errors in 0 and p determinations due to
random fluctuations, are given in section III. Correct measurements of
mass attenuation coefficients are essential to the dual-energy gamma at-
tenuation method. Rewriting equations [Biii] and [Biv] of Appendix B
in matrix form, provides the following mathematical form:
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