Philip A. Townsend, Keith N. Eshleman, and Chris Welcker. 2004. Remote sensing of gypsy moth defoliation to assess variations in stream nitrogen concentrations. Ecological Applications 14:504–516.

Appendix A. Computation of change vector statistics using Tasselled Cap brightness (B), greenness (G), and wetness (W) for images from two dates. A pdf file is also available.

The equations to compute change vector statistics using Tasselled Cap brightness (B), greenness (G), and wetness (W) for images from two dates are:

(1) Magnitude change, , where B, G, and W refer to the changes in image BGW between the two dates,

(2) Colatitude, , and

(3) Longitude,

To compute the summary spherical statistics (such as the mean ¸ or   for all for all forested pixels in a sub-basin), the direction cosines x, y, and z are computed for the B, G, and W dimensions for every pixel:

(4) ,  and .

These values are summed for all forested image pixels i in a sub-basin such that mean direction cosines are calculated as:

(5)
with in the numerator of  and in the numerator of .

Finally, the mean direction angles are:
 
(6) , and
(7) .
Equations 1–4 follow standard trigonometry and were derived from CRC Standard Tables and Formulae (Zwillinger 1995).  Our application of change vector statistics follows from Allen and Kupfer (2000), but differs in implementation with Eqs. 2 and 7.

Literature cited

Allen, T. R., and J. A. Kupfer. 2000. Application of spherical statistics to change vector analysis of Landsat data: Southern Appalachian spruce-fir forests. Remote Sensing of Environment 74:482–493.

Fisher, N. I., T. Lewis, and B. J. J. Embleton. 1987. Statistical analysis of spherical data. Cambridge University Press, Cambridge.

Zwillinger, D., editor. 1995. CRC Standard Mathematical Tables and Formulae, 30th Edition. CRC Press, Boca Raton, Florida, USA.