# Transfer Functions for Predicting Borehole Gamma-Ray Logs

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## Abstract

If the gamma-ray emission rate of rocks varies only with depth, the intensity recorded in a vertical borehole is mathematically predictable. The Fourier transforms of these two functions of depth relate multiplicatively via transfer functions expressible as infinite series whose terms depend on Bessel functions. The results encode readily as programs with imperceptible run times that sample the transfers regularly for use with digitized data. Separate transfers treat non-directional and directional detectors, both supposed of negligible size and located on the borehole axis. Predicting the log from a synthetic test bed confirms and illustrates the theory.

## Keywords

Borehole Fourier transform gamma-ray transfer function## Mathematics Subject Classification

86-08## Notes

### Acknowledgements

Dr. Bruce Dickson initiated the work reported here, as well as in Dickson and Craig (2012), by a 16 July, 2009 email asking whether the borehole case might have an analytic solution like the airborne one (Craig 1993). Both he and Mr. Geoff Taylor greatly assisted my endeavours to find answers.

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