Maximum Likelihood Spectral Fitting: The Batchelor Spectrum

Abstract

We describe a simple technique for fitting spectra that is applicable to any problem of adjusting a theoretical spectral form to fit observations. 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. This is because the range of wavenumbers used in the fit does not vary, and the built-in noise model forces the routine to ignore the spectrum as it gets down towards the noise level. 3. Error bars. There is a theoretical estimate for the variance of the fitted parameters, based on how broad or narrow the likelihood function is in the vicinity of its peak. 4. We calculate statistical quantities that indicate how well the observed spectrum fits the theoretical form. This is extremely useful in automating analysis software, to get the computer to automatically flag "bad" fits. The method is demonstrated using data from SCAMP (Self-Contained Autonomous Microstructure Profiler), 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 pseudo-data generated by Monte-Carlo techniques. Pseudo-code outlines for the spectral fit routines are given.

Poincaré wave–induced mixing in a large lake

“…Rate of dissipation and vertical diffusivity: The rate of dissipation of turbulent kinetic energy e was computed from the SCAMP data by fitting the measured temperature gradient spectra to the theoretical Batchelor spectrum using 25-cm segments and the maximum-likelihood spectralfitting method developed by Ruddick et al (2000; and recently improved by Steinbuck et al [2009]). These methods provide a good fit to the theoretical Batchelor spectrum and give reliable criteria to reject bad segments.…”

Internal waves downstream of Norfolk Ridge, western Pacific, and their biophysical implications

“…SCAMP is a loose-tether instrument that free falls at 10 cm s 21 recording temperature, conductivity, temperature gradient, and fluorescence at 100 Hz (Ruddick et al 2000). The workboat operated within 1 km of the R/V Tangaroa and profiles were typically located between the vessel and the buoyed DVA.…”

“…The temperature microstructure were divided into 4-m sections, each containing around 4000 data points, which, in turn, were divided into 64-point segments for the spectral fitting. The goodness-of-fit criterion based on a comparison of the observed to theoretical temperature gradient spectrum (Ruddick et al 2000) and a noise threshold were used, so that 45% of e samples were rejected.…”

“…The temperature microstructure data enable calculation of a number of properties such as the rate of turbulent energy dissipation e and vertical diffusivity K r , where the estimate of K r is direct and does not assume a so-called mixing efficiency (Ruddick et al 2000). These were all binned to 10-m resolution, as was the Ri gr (Eq.…”

Estimating turbulent kinetic energy dissipation using the inertial subrange method in environmental flows

“…To avoid these issues, we chose the maximum likelihood estimator (MLE) technique for spectral fitting (Priestley 1981;Bendat and Piersol 2000). Ruddick et al (2000) successfully applied the MLE technique to Batchelor fitting of temperature gradient spectra. The MLE for n samples is: (12) where f(a i ) is the probability density function of the i th observation of variable a, which for our application is the estimated velocity spectra .…”