The main goal of GSO: given an unknown naturalgamma ray spectrum X = x1, x2, ...xn, compute its K, U, and Th fractional content. (Further computations and calibrations are then required to compute%K andppm U and Th.) Matrix inversions in realtime are avoided by this GSO method and total realtime computations are minimized.
Shown below are RAW and NORMALIZED standard spectra for K, U, and Th, as well as theirassociated Gram-Schmidt spectral vectors e1, e2, e3. All these vectors are computedoffline in a laboratory computer environment.
RAW K, U, Th standard spectra are typically acquired at special Test Pits located atthe 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. Computee1*e1 (1.0e06), e2*e2, and e3*e3; compute e1*Knorm, e1*Unorm, ande2*Unorm.
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 NORMALIZEDX 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 contentof the unknown spectrum X using simple algebraic expressions. In the presentscheme with e1 chosen as Th, the U fractional content calculation is simplest and the Thcontent is the most complex.