SPECPR Users’ Manual                                            Page 8-f42.1

F42:  Fit Band Profile from a Reference Spectrum        Alias = bandmap
This routine fits an absorption band from a reference spectrum to an
observed spectrum by minimizing the least squares.  The reference
absorption band depth is changed by a simple equation (thus the
reference spectrum chosen should have an absorption band profile
similar in saturation level to the observed spectrum in order to
provide the best fit).
This routine is described in detail in the paper

Clark, R.N., A.J. Gallagher, and G.A. Swayze:
Material Absorption Band Depth Mapping of Imaging Spectrometer Data
Using a Complete Band Shape Least-Squares Fit
with Library Reference Spectra,
Fifth Airborne Imaging Spectrometer Workshop Proceedings,
JPL Publication 90-54, 176-186, 1990.

To use f42, the user types the observed spectrum and f42 from the
math command line.  For example, if v76 is the observed spectrum,
you would type:

             v76f42

Next you will be prompted to enter the reference library spectrum
file ID and record number.

Then you are asked to enter the left and right continuum values.
The continuum is located on each side of the absorption band and
consists of one or more data channels.  You enter the beginning and
ending channels for each side.  For example, if the channels
describing the left side of the continuum included channels 331,
332, and 333 and if the right side continuum included channels 385,
386, 387, and 388, you would enter the following.

             331 333 385 388

If the continuum included only one channel, you still must enter 4
numbers, so the beginning and ending channel would be the same.  For
example if the continuum was only channel 331 on the left and 339
and 340 on the right, you would type:

             331 331 339 340

Next you are asked what data to output to the file.  You can save
the fitted reference spectrum (press return, this is the default),
or the continuum-removed observed spectrum by typing "o" (you don’t
type the quotes).

The Band depth Fitting Algorithm
The absorption band depth, D, is defined relative to its continuum:

          D = 1 - Rb/Rc                                         (eqn 8.f42.1)

where Rc is the reflectance of the continuum at the band center, and Rb is the reflectance at the band center. The definition originates from Clark and Roush, (JGR, 1984). The process of absorption band analysis is to first remove a continuum (see Clark and Roush, 1984 for details about continuum analysis). Once the continuum is removed, a band can be characterized by determining how well the feature matches a reference library spectrum. A simple model to change a continuum-removed absorption feature’s contrast is to add a constant to the data at all wavelengths. In this case, the feature will not be represented properly if the band saturation changed significantly, due to say a major change in grain size between the reference and observed spectra. The algorithm presented here uses the simple case of an additive component because it is computationally fast compared to a full radiative transfer model. The method can easily be adapted to the full model however, or even to a method that examines the band depth and chooses an appropriately saturated absorption feature from a library of materials at several grain sizes. The algorithm first removes a straight line continuum from the library reference spectrum using channels on each side of the absorption feature that is to be mapped. The continuum is removed from the observed spectrum using the same method and spectral channels. The user may select several channels on each side of the band so that noise in the continuum is averaged. The continuum is removed by division:

              L(λ)                       O(λ)
      Lc(λ) = —————    and       Oc(λ) = —————,         (eqn 8.f42.2 a and b)
              Cl(λ)                      Co(λ)

where L(λ) is the library spectrum as a function of wavelength, λ, O is the observed spectrum, Cl is the continuum for the library spectrum, Co is the continuum for the observed spectrum, Lc is the continuum removed library spectrum, and Oc is the continuum removed observed spectrum. The contrast in the reference library spectrum absorption feature is modified by a simple additive constant, k:

                     Lc + k
               Lc’ = ——————,                                    (eqn 8.f42.3)
                     1.0 +k

where Lc’ is the modified continuum-removed spectrum that best matches the observed spectrum. This equation can be rewritten in the form:

              Lc’ = a + bLc,                                    (eqn 8.f42.4)

where

             a =  k /(1.0 + k), and

             b = 1.0/(1.0 + k).                                (eqn 8.f42.5)

In Equation 8.f42.4 we want to find the a and b that gives a best fit to the observed spectrum Oc. The solution is done using standard least squares:

              a = (Σ Oc - bΣ Lc)/n,

                 Σ OcLc - Σ OcΣ Lc/n
              b = ———————————————————,
                  Σ Lc2 - (Σ Lc)2/n

and
              k = (1-b)/b,                                      (eqn 8.f42.6)

where n is the number of spectral channels in the fit. The algorithm produces a band depth, indicating its spectral abundance in the observed image, a goodness of fit (correlation coefficient) which gives a measure of confidence in the accuracy of the resulting fit, and the continuum value at the band center in the observed spectrum. The correlation coefficient is computed by

              r = sqrt(b1b’1)                             (eqn 8.f42.7)

where b1 is from Bevington (Data Reduction and Error Analysis for the Physical Sciences, Mcgraw Hill, New York, 1969), his equation 7.2 (which he calls b) on page 120, and b’1 is from Bevington (1969), his equation 7.3 (which he calls b’) on page 121.