Appendix A. Mathematical equations for soil water content
 (0) and bulk density (p) determinations using
 the monoenergetic gamma-ray attenuation method
 For the monoenergetic gamma-ray attenuation method, equation [4]
 can be rewritten to describe attenuation of collimated monoenergetic
 gamma radiation in soil as
 I = I, exp { f-/ ps X I-w pw X,, /La pa Xa 2c,, pc xn
 -a, pa (x,, + Xa2)} I[Ai]
 where the subscripts s, w, a, and c represent soil, water, air, and the wall
 of the container, respectively. The distances from the radiation source
 to the soil column and from the soil column to the radiation detector are
 represented as x,L and xa2. The third and fifth terms in the exponent of
 equation [Ai] are insignificantly small compared to the remaining terms
 and can thus be excluded. During routine measurements of the intensity
 of the gamma radiation, the incident radiation, Io, is commonly ob-
 tained after the radiation beam has passed through the empty soil con-
 tainer. Thus the fourth term of the exponent of equation [Ai] which
 represents the attenuating influence of the container can be eliminated
 by rewriting the equation in a simpler form as
 I =Io exp { -js ps Xs --w ptw xw} [Aii]
where x, is the equivalent thickness of the soil and x,, is the equivalent
thickness of water in the bulk soil. The thickness of the bulk soil, x, is
thus the sum of x,, x,,, and x, as shown in Fig. 1, or expressed in other
words, the sum of the volumetric fractions of soil, 0,, water, 0, and air,
0G, for the bulk soil is unity. Since the product of the particle density
(ps) for soil and the equivalent soil thickness (x,) is the same as the
product of the soil bulk density (p) and the thickness of the bulk soil
sample (x), and since the product of the water density (pw) and the
equivalent water thickness (x,,,) equals the product of 0 and x, equation
[Aii] can be further simplified as
 I = Io exp { -x(,sp + p,, 9)) [Aiii]
Thus attenuation of a collimated beam of monoenergetic gamma rays
during transmission through a soil column can be attributed to the com-
bined effects of the bulk density, p and the volumetric water content, 0,
of the soil.
 Assuming p to be invariant with time and using carefully measured
values for the parameters x, /, and f,, equation [Aiii] can be rearranged
to obtain the water content of the soil

 0 = [--1 In ( +x) p [Aiv]

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