94.

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Page 182

(4.46)

where ∂ indicates the partial derivative.

It can be inferred that the linear mixture model is more appropriate when the number of input features k is reasonably large, certainly larger than the seven-band case of Landsat TM. For remote sensing data with a small number of spectral bands such as the three-band SPOT HRV, the linear mixture model is unsuitable because only a very small number of end members can be specified. There is, however, a solution which can overcome the restriction of the number of bands. We will be concerned with this method in later paragraphs. Linear spectral unmixing is more successfully applied to hyperspectral data, in which the number of bands is no longer a constraint. However, inter-correlation between the bands of the hyperspectral data set can pose numerical problems in terms of solving the set of linear equations. An orthogonal transformation such as the MNF (Chapter 2) can help by removing the effect of the correlation and at the same time minimising the noise level of the data.

Before a mixture model is constructed, we need to determine the constants, i.e. the reflectances of the end members (a_i and b_i above). Normally, training data are used to obtain those constants. The training data consists of a set of ‘pure’ pixels representing the end members. These training data can be extracted from the image, or may be derived from laboratory measurements (Smith et al., 1980; Asner and Lobell, 2000). The training process has to be performed carefully, or proportions greater than 1.0 or less than 0.0 may result. For example, if a pixel containing 80 tree cover is selected to represent the end member ‘trees’, then any pixel containing a proportion of trees greater than 80% will receive a fractional proportion greater than 1.0. A similar argument can be used to show that negative proportions are also possible if the training data do not represent pure pixels. Proportions greater than 1.0 or less than 0.0 are known as over-shoots and undershoots, respectively. Their presence indicates that the simple linear mixing model is invalid. Either non-linear mixing is taking place or a significant end member has been omitted.

As noted earlier, the number of end members is limited by the number of available spectral bands or features. In general, multispectral image sets with a small number of bands do not produce adequate results when linear spectral unmixing is applied (van der Meer, 1995). Even where the number of bands is sufficiently large, the application of the linear mixture model should be carried out carefully because some combinations of spectral bands may be highly correlated, as noted above. These high correlations can cause computational problems in solving the least-squares equations, for high correlations imply that the true dimensionality of the data is less than the number of bands. Simply increasing the number of bands in linear

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