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Maximum Likelihood Spectral Fitting: The Batchelor Spectrum

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  • 1 Department of Oceanography, Dalhousie University, Halifax, Nova Scotia, Canada
  • | 2 Department of Oceanography, Dalhousie University, Halifax, Nova Scotia, Canada, and Israel Oceanographic and Limnological Research, Haifa, Israel
  • | 3 Department of Oceanography, Dalhousie University, Halifax, Nova Scotia, Canada
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Abstract

A simple technique for fitting spectra that is applicable to any problem of adjusting a theoretical spectral form to fit observations is described. All one needs is a functional form for the theoretical spectrum and an estimate for the instrumental noise spectrum. The method, based on direct application of the maximum likelihood approach, has several advantages over other fitting techniques. 1) It is unbiased in comparison with other least squares or cost function–based approaches. 2) It is insensitive to dips and wiggles in the spectrum, due to the small number of fitted parameters. It is also robust because the range of wavenumbers used in the fit is held fixed, and the built-in noise model forces the routine to ignore the spectrum as it gets down toward the noise level. 3) The method provides a theoretical estimate for error bars on the fitted Batchelor wavenumber, based on how broad or narrow the likelihood function is in the vicinity of its peak. 4) Statistical quantities that indicate how well the observed spectrum fits the theoretical form are calculated. This is extremely useful in automating analysis software, to get the computer to automatically flag “bad” fits.

The method is demonstrated using data from the Self-Contained Autonomous Microstructure Profiler (SCAMP), a free-falling temperature microstructure profiler. Maximum likelihood fits to the Batchelor spectrum are compared to the SCAMP-generated fits and other least squares techniques, and also tested against pseudodata generated by Monte Carlo techniques.

Pseudocode outlines for the spectral fit routines are given.

Corresponding author address: Dr. Barry R. Ruddick, Department of Oceanography, Dalhousie University, Halifax, NS B3H 4J1, Canada.

Email: barry.ruddick@dal.ca

Abstract

A simple technique for fitting spectra that is applicable to any problem of adjusting a theoretical spectral form to fit observations is described. All one needs is a functional form for the theoretical spectrum and an estimate for the instrumental noise spectrum. The method, based on direct application of the maximum likelihood approach, has several advantages over other fitting techniques. 1) It is unbiased in comparison with other least squares or cost function–based approaches. 2) It is insensitive to dips and wiggles in the spectrum, due to the small number of fitted parameters. It is also robust because the range of wavenumbers used in the fit is held fixed, and the built-in noise model forces the routine to ignore the spectrum as it gets down toward the noise level. 3) The method provides a theoretical estimate for error bars on the fitted Batchelor wavenumber, based on how broad or narrow the likelihood function is in the vicinity of its peak. 4) Statistical quantities that indicate how well the observed spectrum fits the theoretical form are calculated. This is extremely useful in automating analysis software, to get the computer to automatically flag “bad” fits.

The method is demonstrated using data from the Self-Contained Autonomous Microstructure Profiler (SCAMP), a free-falling temperature microstructure profiler. Maximum likelihood fits to the Batchelor spectrum are compared to the SCAMP-generated fits and other least squares techniques, and also tested against pseudodata generated by Monte Carlo techniques.

Pseudocode outlines for the spectral fit routines are given.

Corresponding author address: Dr. Barry R. Ruddick, Department of Oceanography, Dalhousie University, Halifax, NS B3H 4J1, Canada.

Email: barry.ruddick@dal.ca

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