roughly dictated by the correlations between them. Soils with high Pb, F, and OM, for 

 example, would not occur. Extension of expectation over these areas is simply a sacri- 

 fice to ease of graphic display. A tendency toward convergence of surfaces for all 

 levels of percent fines (F) was expected to occur with decreasing organic matter (OM) . 

 With increasing OM and when approaching extremes of bulk density (Pb) , the suppressive 

 effects of decreasing F on retention were expected to be accentuated. 



Model Development 



The position of the data array in the expected model and a more precise estimate of 

 the nature of the interaction was next ascertained. Expected retention trends over OM 

 were fitted to the plotted data points by approximate least deviations (for both 1/3 

 and 15 bars), for two or three F-strata, within each of three Pb-strata (see fig. 5). 



It was apparent from these data trends that the data lay in the right front quarter 

 of the expected surfaces and were fairly well distributed over low to medium F with only 

 a few high F-values being present. Furthermore, the expected interactions appeared to 

 exist; thus, our next effort was directed at generating a suitably accurate interaction 

 descriptor. Using such a model, we hoped to minimize curve form bias likely to be 

 associated with the application of less sensitive transformations. 



The curves over OM were characterized using algebraic forms identified with Matcha- 

 curves-1 and -2 (Jensen and Homeyer 1970 and 1971) . Differences in these functions 

 (intercepts, maxima, and degree of curvature), smoothed in accord with expectation, were 

 then expressed as appropriate Matchacurve functions of Pb and F to arrive at an algebraic 

 approximation of the four-dimensional relation. And finally, the algebraic function was 

 refitted to the original data points by least squares, using a simple linear model 

 having zero intercept. 



The complexity of the final descriptors probably will appear objectionable in our 

 old frame of reference where the lack of available computers necessitated descriptor 

 simplicity. This is no longer true, and where complex algebra was required to describe 

 particular main effects and interactions in the model here, we were not reluctant to 

 use it. The descriptors herein were developed over a period of 1 week and final scal- 

 ing passes in the computer were extremely short and inexpensive. 



1.08 1.28 1.53 



Pb 



Figure o. --Expected trends fitted to the data. 



6 



