The Virginia Beach data were analyzed at the Northwestern University 
Computing Center with an IBM 709. The program computes a (U,V) matrix and 
a series of vectors for each variable. The matrix is inverted and post- 
multiplied by the vectors to obtain a set of polynomial coefficients. 
These in turn are used to compute the trend-surface values. Some of the 
underlying theory and a description of the program are given by Krumbein 
(1959). Whitten (1963) describes the program in detail. 
When a trend surface is fitted to map data, a computed value is 
obtained for each control point. These are plotted on a base map and 
contoured by hand; alternatively, the computer generates a field of com- 
puted values and prepares a trend map either directly or through use of an 
X-Y plotter. The computed trend value for a given map point commonly does 
not agree exactly with the observed value, so that a deviation (which may 
be positive or negative) is associated with each map point. Thus, if X is 
the observed value of the mapped variable at point (U,V), and if X' is the 
computed trend value, then the deviation from the fitted surface is defined 
as R = X - X'. These deviations may also be mapped to see what the varia- 
tion pattern is after one or more low-order trend surfaces have been ex- 
tracted. Deviation maps are commonly contoured by hand; and, because some 
freedom in drawing contours is usually present, the deviation maps presented 
here were all prepared by linear interpolation between a given point and 
surrounding points. 
Linear Surface. The linear surface is fitted to irregularly spaced 
map data by setting up the relation: 
ak = a S Gy oR) 
where X is the observed value at point (U,V); A; is a constant that 
represents the "height" of the trend surface at (U,V) = (0,0); and Br and 
Cr are the coefficients of the linear surface. The deviation, R,, contains 
trend components higher than linear, plus some unknown content of non- 
systematic, seemingly random fluctuations. 
The computed linear trend value, X;', is obtained from the relation 
Xy' = Ay + BLU + CLV by multiplying the U-coordinate of a given point by 
BL, multiplying its V-coordinate by C;, and adding A; to the sum of the 
two products. Rp, as stated, is simply the numerical difference between 
the observed and computed values at each point of observation. 
Quadratic Surface. The complete quadratic surface includes two 
linear and three quadratic terms, and is fitted by setting up the relation: 
X = AQ + BQU + CQv + DQu2 + EQUV + Fv? + RQ 
Here, X is again the observed value of the mapped variable at a given (U,V) 
point, Ag is a constant, and Bg, Cq, --., Fa are the polynomial coefficients 
