of the quadratic surface. The quadratic is computed as a whole because 
the numerical values of the constant and of the coefficients change as 
higher surfaces are considered, as shown by the change in subscripts from 
IL, UO @. 
The remainder RQ now contains trend components higher than quadratic 
plus purely random fluctuations. As before, if Xg' is the computed value 
of the quadratic surface at a given control point, then Xo) 26 = OKO)" 
Cubic Surface. The complete cubic trend contains two linear, three 
quadratic, and four cubic terms, as well as a constant A. Inasmuch as 
the numerical values of the coefficients again change, the coefficients 
on the cubic surface are subscripted as shown: 
a eB CUNHCcVale DCUCRA NM ECUVE HaREViCN MAGES) A pHEUCV) on JeuUV =) Ke Viole 
The deviations on the cubic surface, R,~, contain trend terms higher than 
cubic, as well as some content of random variations. Our analysis did not 
extend beyond the cubic surface. 
Sum of Squares Evaluation of Trend-Surface Maps. The total sum of 
squares of a mapped variable is a measure of the variability of map data. 
It is expressed as the sum of the squared differences of the observed 
valués from their mean: 
zt Tae 
SSy = (X = X) 
The expression represents the sum of the squared deviations of X from 
their mean value, X, at all control points. The sum of squares of the 
deviations is computed as: 
SSp = R2 
since the sum of R over the whole map is zero. 
The sum of squares associated with the computed values, X', can be 
expressed as: 
where X is the mean of the observed X; in trend analysis the computed mean, 
X', is the same as the mean of the observed data, X. 
The "strength" of a trend surface can be evaluated informally by 
noting how much of the total map variability is "accounted for" by the 
fitted surface. This is computed as 100(SSx'!/SSx). 
