Gram-Schmidt Orthonormalization (GSO)
The main goal of GSO: given an unknown natural gamma ray spectrum X = x1, x2, ...xn, compute its K, U, and Th fractional content. (Further computations and calibrations are then required to compute %K and ppm U and Th.) Matrix inversions in realtime are avoided by this GSO method and total realtime computations are minimized.
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Shown below are RAW and NORMALIZED standard spectra for K, U, and Th, as well as their associated Gram-Schmidt spectral vectors e1, e2, e3. All these vectors are computed offline in a laboratory computer environment.
RAW K, U, Th standard spectra are typically acquired at special Test Pits located at the University of Houston. These NORMALIZED standard spectra each have an (arbitrary) dot product of 1.0e06. The Gram-Schmidt vectors are constructed from these standard NORMALIZED K, U, Th spectra. In this example, the first Gram-Schmidt vector e1 was chosen as the NORMALIZED Th spectrum. These Gram-Schmidt vectors are mutually perpendicular. Compute e1*e1 (1.0e06), e2*e2, and e3*e3; compute e1*Knorm, e1*Unorm, and e2*Unorm.
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Once an unknown spectrum X is acquired in realtime, it is normalized so that
its own dot product is also 1.0e06. Dot products are formed with this NORMALIZED X spectrum and the Gram-Schmidt vectors e1, e2, and e3.
Finally, use all the a priori computed dot products to compute the K, U, and Th fractional content of the unknown spectrum X using simple algebraic expressions. In the present scheme with e1 chosen as Th, the U fractional content calculation is simplest and the Th content is the most complex.
