# Full text: Mesures physiques et signatures en télédétection

```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)]
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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 >.
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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
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```

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