NORMAL GEOTHERMAL GRADIENT gI 
Equations 7 and 8 are the same as equations 2 and 3. Since the 
weights vary inversely as the squares of the probable errors, it follows 
that in equations 2 and 7, the errors of the observed temperatures 
are proportional to the depths, whereas in equations 3 and 8 the 
errors are proportional to the square roots of the depths. Equation 9 
amounts to assuming that the errors in the temperature readings 
at the various depths are the same. In 10, the errors are assumed to 
vary inversely as the square root of the depth, that is, the errors of 
the observed temperatures are supposed to diminish with the depth. 
There is no justification for this assumption, but equations 7, 8, 9, 
in which the errors are supposed to be independent of the depth 9, 
or to increase linearly with the depth 7, or the square root of the depth 
8, rest on a sufficiently sound basis to justify their use.. Equation 6 is 
a partial check on the calculations. 
When the observations in the different wells are made at the 
same depth (21), equations 7, 8, 9, 10 reduce to 
ae yi/x1 + ye/ x2 + CEG AG Vn/%n 
n 
the arithmetic mean of the gradients. Equation 6 shows that the 
sum of the residuals vanishes for these special cases. That is, 
Ute Ue aio Ino 
Formulas 7, 8, 9 are applicable to the data in the last column 
of Table I. In the remaining tabulations of 6 in Table I, the values 
have been determined from a series of n’ observed temperatures in 
the same well by making a least-square solution of the m’ observation 
equations: 
a+txmb=y 
b= 
a+ %2b = Ye @) 
a+ xn'b = yn’ 
where y is now the observed temperature at depth x; a is the intercept 
of the straight line on the y-axis; and 6 is the computed gradient. 
The least-square solution of equations 11 gives the normal equa- 
tions: 
n'a + (Zx)b = Ly 
(2x)a + (2x)2b = Day 
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