351 
so that <p has a mean value of one within the integration interval. At each 
within the wavelength interval dA. . Thus successive residuals 6x^ pass 
through zero at more and more wavelength values as the order of the expansion 
Illustrations of basis functions may be found in Price, 1990 and 1992. 
The criterion for terminating the selection of spectral bands is based on the 
comparison of the residuals with the noise present in the ensemble of spectra. 
Let percent error E be defined by 
where the integral extends over the wavelength range of interest, excluding 
the regions of strong water vapor absorption in the atmosphere at 1.35 to 1.47 
/am and 1.81 to 2.02 /am, since these are not usable for airborne and satellite 
observations. For terminating the expansion in basis functions we generally 
require E(M) < 0.01%, or less for high signal to noise data, as, for example 
laboratory spectra. This value 0.01% corresponds to the noise level at a 
measurement signal to noise ratio of 100:1, e. g. to a mean square reflectance 
error of 0.2% at 20% reflectance. However while some collections of 
laboratory and field reflectance spectra are relatively noisy, others are 
nearly free of random variations at the 0.01pm scale. Also some data sets 
have higher noise at the upper and lower ranges of the observed spectrum and 
lower noise in the midrange (0.6-1.3pm). Thus a simple condition on residuals 
is not sufficient to guarantee that all spectral features are described. 
3. APPLICATION TO SURFACE AND LABORATORY SPECTRA 
When carrying out statistical processing one must utilize as general a data 
set as possible in order to include variability reasonably expected in 
satellite or aircraft data. For this analysis a number of collections of 
spectra have been studied, including soils, vegetation, and igneous and 
sedimentary rocks. Many of these data sets have become available only 
recently. The first two data sets do not span the full range 0.40-2.50 pm 
like the others, but they are the most complete spectral data sets available 
for describing conventional agriculture. We describe the data sets briefly: 1 
1. Agricultural crops. This collection (Biehl, et al., 1984) contains 
approximately 1400 field spectra from soybeans, corn, and winter wheat, with a 
few observations of sunflowers and alfalfa and bare soil, taken throughout 
several growing seasons. In the era of these measurements field spectrometers 
were less advanced. The useable interval was 0.50-2.31 pm, with the water 
vapor intervals (1.35-1.47 and 1.80-2.02 pm ) deleted due to low signal. 
These spectra have been discussed previously (Price, 1990, 1992). During this 
analysis it was found by comparison with vegetation spectra from the other 
collec tions that some data from 1978 were corrupted by drift of one of the two 
detectors due to temperature changes. (C. Daughtry and L. Biehl, private 
communication, 1992). Elimination of these spectra left a total of 1276. 
iteration level i the residual vector 5x^ is approximated by Sx^ = <jn(A) 
Ct 
leaving a new residual 5x^ + ^. Then the procedure moves to 5x^ + ^. From the 
definition 5x. and all higher order residuals have the value zero somewhere 
A 
increases, and the magnitude of the residuals J (Sx) dA decreases. 
E(M) = 100% 
(5)