^{1,a)}and J. Tinsley

^{1}

### Abstract

The covariance method exploits fluctuations in signals to recover information encoded in correlations which are usually lost when signal averaging occurs. In nuclear spectroscopy it can be regarded as a generalization of the coincidence technique. The method can be used to extract signal from uncorrelated noise, to separate overlapping spectral peaks, to identify escape peaks, to reconstruct spectra from Compton continua, and to generate secondary spectral fingerprints. We discuss a few statistical considerations of the covariance method and present experimental examples of its use in gamma spectroscopy.

We wish to thank Ray Keegan, Bill Quam, Howard Bender, and John Di Benedetto for their many suggestions and insights. The work was performed under the auspices of the U.S. Department of Energy (DOE), National Nuclear Security Administration by National Security Technologies, LLC, under Contract No. DE-AC52-06NA25946. Financial support was provided by the U.S. DOE, Office of Defense Nuclear Nonproliferation Research and Development (NA-22). The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript or allow others to do so, for United States Government purposes.

I. INTRODUCTION

II. COVARIANCE AND LATENT VARIABLE EXTRACTION

III. THE CORRELATED POISSON DISTRIBUTION

IV. PRECISION OF COVARIANCE AND COINCIDENCE ESTIMATES

V. APPLICATIONS

A. Noise filtering

B. Spectral fingerprints

C. Separation of overlapping variables

D. Compton reconstruction

E. Escape peak identification

VI. CONCLUSION

### Key Topics

- Probability theory
- 21.0
- Compton scattering
- 4.0
- Data acquisition
- 4.0
- Sodium
- 4.0
- Clocks
- 3.0

##### H01L31/115

##### H04N5/32

##### H05G

## Figures

The comparison of the precisions estimates of (a) covariance and (b) coincidence for covariances ranging from three orders of magnitude below the noise terms to three orders of magnitude above the noise. The noise terms have been set to 1. In all cases the precision of the covariance is better than that of the coincidence.

The comparison of the precisions estimates of (a) covariance and (b) coincidence for covariances ranging from three orders of magnitude below the noise terms to three orders of magnitude above the noise. The noise terms have been set to 1. In all cases the precision of the covariance is better than that of the coincidence.

Scaling of the resolving time shows how covariance can extract correlated signal even when accidental coincidences overwhelm correlated coincidences. Curve (a) is the coincidence signal. Curve (b) is the estimated accidental coincidence contribution, and curve (c) is the covariance. On these time scales the correlated signal is constant, so the true coincidence curve, represented by the covariance, should remain flat.

Scaling of the resolving time shows how covariance can extract correlated signal even when accidental coincidences overwhelm correlated coincidences. Curve (a) is the coincidence signal. Curve (b) is the estimated accidental coincidence contribution, and curve (c) is the covariance. On these time scales the correlated signal is constant, so the true coincidence curve, represented by the covariance, should remain flat.

The positive covariances in this map show correlations between cascade gammas emitted after the radioactive decay ^{133}Ba. Such correlations are nuclide specific and can serve as secondary spectral signatures.

The positive covariances in this map show correlations between cascade gammas emitted after the radioactive decay ^{133}Ba. Such correlations are nuclide specific and can serve as secondary spectral signatures.

Negative covariances in this map shows competing pathways in the relaxation of the ^{133}Cs daughter nucleus from radioactive decay of ^{133}Ba. These features can also be exploited for spectral identification. This map was generated from the same data that produced the map shown in Fig. 3 .

Negative covariances in this map shows competing pathways in the relaxation of the ^{133}Cs daughter nucleus from radioactive decay of ^{133}Ba. These features can also be exploited for spectral identification. This map was generated from the same data that produced the map shown in Fig. 3 .

This covariance map shows three islands of covariance in the vicinity of 1.3 MeV. The data are from a ^{60}Co source masked by a stronger ^{22}Na source. The average spectra show only two peaks, but the covariance map can resolve three. The features marked (a) and (b) are from the ^{60}Co correlated cascade gammas at 1.17 MeV and 1.33 MeV. The feature marked (c) is the ^{22}Na correlation between 511 keV and 1.27 MeV.

This covariance map shows three islands of covariance in the vicinity of 1.3 MeV. The data are from a ^{60}Co source masked by a stronger ^{22}Na source. The average spectra show only two peaks, but the covariance map can resolve three. The features marked (a) and (b) are from the ^{60}Co correlated cascade gammas at 1.17 MeV and 1.33 MeV. The feature marked (c) is the ^{22}Na correlation between 511 keV and 1.27 MeV.

This covariance map is for stand-off detection by a NaI detector array of a ^{60}Co source. No covariance features from source correlations are visible, and all the features seen here are from detector crosstalk. Compton reconstruction of this covariance table produces the spectrum shown in Fig. 7 .

This covariance map is for stand-off detection by a NaI detector array of a ^{60}Co source. No covariance features from source correlations are visible, and all the features seen here are from detector crosstalk. Compton reconstruction of this covariance table produces the spectrum shown in Fig. 7 .

Spectral reconstruction of Compton crosstalk is accomplished by cross diagonal summing of the covariance table. Curve (a) is reconstructed spectra from the Compton continuum by processing the covariance table for the data of Fig. 6 . Shown for comparison, the curve labeled (b) is the average spectra. Reconstruction of the Compton continuum of these data yields 50% extra photo peak signal.

Spectral reconstruction of Compton crosstalk is accomplished by cross diagonal summing of the covariance table. Curve (a) is reconstructed spectra from the Compton continuum by processing the covariance table for the data of Fig. 6 . Shown for comparison, the curve labeled (b) is the average spectra. Reconstruction of the Compton continuum of these data yields 50% extra photo peak signal.

The covariance filtered (a) background spectrum from an array of HPGe detectors shows peaks that are correlated with the 511 keV gamma peak from pair production. Single and double escape peaks of ^{209}Tl are located at 2103 keV and 1592 keV, and the single escape peak of ^{40}K is visible at 950 keV. The negative covariance at 1460 keV indicates an anti-correlation between pair production and the full energy photo peak of ^{40}K. For reference, the average spectrum (b) is plotted scaled and offset above the filtered spectrum.

The covariance filtered (a) background spectrum from an array of HPGe detectors shows peaks that are correlated with the 511 keV gamma peak from pair production. Single and double escape peaks of ^{209}Tl are located at 2103 keV and 1592 keV, and the single escape peak of ^{40}K is visible at 950 keV. The negative covariance at 1460 keV indicates an anti-correlation between pair production and the full energy photo peak of ^{40}K. For reference, the average spectrum (b) is plotted scaled and offset above the filtered spectrum.

## Tables

Fourfold table of probabilities for detection of the latent variable *A*.

Fourfold table of probabilities for detection of the latent variable *A*.

First four raw and central moments for a Poisson distribution.

First four raw and central moments for a Poisson distribution.

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