350 
x a = I s“*.(A), 
i=l 
ci» 80 tha 
itérât 
where the basis functions <p(A^, X ^, A^, ...A ) are spectral shapes, as defined 
from statistical analysis of a collection of spectra, and the coefficients S a 
i 
are wavelength integrals over specific band limits for the original 
measurements x a . Each basis function tp^ has an associated spectral interval 
[A^Cmin), A^Cmax)] representing the domain of integration for determining the 
coefficients S^. Thus each <p has essentially unit value in its spectral 
interval (more precisely has mean value of 1.0 in this interval), then 
decreases according to the degree of wavelength correlation in the ensemble of 
measured spectra. The expansion represents successive approximations to the 
original spectra. Evidently the number of basis functions M which is required 
to describe the x a to within very small residuals must be much less than the 
spectral dimensionality n, or else the expansion is not useful. In this 
section we describe first the definition and properties of the expansion, 
where the formalism for obtaining a set of basis functions <p by a single set 
of computations has been described in Price, 1993. 
2.1 Definition and Properties of the Expansion 
From inspection we know that most visible to near infrared reflectance spectra 
vary in a relatively smooth fashion, implying that correlations exist between 
measurements at nearby wavelengths. Thus a measurement in a limited spectral 
range provides information about values over a interval. Let Sx? be the 
difference between a measured spectrum and the sum through term i of the 
expansion, and S? be the average of Sx^ over the interval [A^(min), A^(max)] 
leavin 
defini 
within 
throug 
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Illust 
The cr 
compar 
Let pe 
E 
where 
the re 
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labora 
nearly 
have h 
lower 
is not 
ST 
l 
1 
[A.(max)-A.(min)1 
! r 
f \ .(max) 
J 1 
} A.(min) 
l 
5x7 dA 
( 2 ) 
At the beginning 5x^ = x. Because the value of is correlated with the 
value of Sx over a wider range, we define the basis function <jcn by 
<p.(A) = < Sx.S. > / < (S.) 2 >. 
l l l / l 
(3) 
3. API 
When c 
set as 
satell 
spectr 
sedime 
recent 
like t 
for de 
where the brackets represent an ensemble average, e. g. < x > = 1/N £ x Q . 
a=l 
From the definition of S, the normalization of <p is 
l |-A^(max) 
i <" x >-V" i „>] A. (min) ” i<X> 
1 
/-a i, max; 
TT —7 7—:—-,—:—rv < Sx. S. > / < (S.)‘ 
[A (max)-A (mm) J , . . i l ' l 
l l A. (mm) 
[A 
. (max) 
> dA 
(4) 
1. Ag 
approx 
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severs 
were 1 
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commur