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B. Ruddick, David Walsh, N. Oakey (1997)
Variations in Apparent Mixing Efficiency in the North Atlantic Central WaterJournal of Physical Oceanography, 27
Compute and upper/lower bounds on from k B results in lines 9 and 11
Compute Y ϭ S obs /S th at each k, for k B of best fit
compute var(Y) ϭ var(S obs /S th ) summed over all wavenumbers
O. Kocsis, H. Prandke, A. Stips, A. Simon, A. Wüest (1999)
Comparison of dissipation of turbulent kinetic energy determined from shear and temperature microstructureJournal of Marine Systems, 21
H. Charnock (1972)
Oceanic Fine StructureNature, 239
W. Press, S. Teukolsky, W. Vetterling, B. Flannery (2002)
Numerical recipes in C
Compute integrated S-N ratio from lines 1 and 2
Take difference of log(likelihood) from lines 9 and 21. Convert to log 10 to get likelihood ratio
calculate log-likelihood function C11 at each wavenumber [Eq. (18)] (call Chi2pdf transformation function); sum C11 over all wavenumbers
Compute var(Y) [Eq. (23)] over wavenumbers for which S B exceeds S n
G. Batchelor (1959)
Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivityJournal of Fluid Mechanics, 5
Compute using observed spectral variance less noise variance [Eq
G. Ivey, J. Imberger (1991)
On the Nature of Turbulence in a Stratified Fluid. Part I: The Energetics of MixingJournal of Physical Oceanography, 21
T. Osborn, C. Cox (1972)
Oceanic fine structureGeophysical and Astrophysical Fluid Dynamics, 3
Compute location of maximum in C11. This is the best-fit k B
D. Luketina, J. Imberger (2001)
Determining Turbulent Kinetic Energy Dissipation from Batchelor Curve FittingJournal of Atmospheric and Oceanic Technology, 18
G. Jenkins, D. Watts (1968)
Spectral analysis and its applications
Plot best-fit spectrum, noise spectrum, and observed spectrum. Superpose best-fit power law spectrum. Annotate this plot with results from lines 2
N. Oakey (1982)
Determination of the Rate of Dissipation of Turbulent Energy from Simultaneous Temperature and Velocity Shear Microstructure MeasurementsJournal of Physical Oceanography, 12
(1954)
Problmes de l'analyse spectrale des sries temporelles stationnaires
with same standard deviation and peak. Plot error bars corresponding to line 11
D. Preston (1983)
Spectral Analysis and Time SeriesTechnometrics, 25
this paper for MLE spectral fitting, and also for computing Batchelor spectra, may be requested directly from the authors
T. Osborn (1980)
Estimates of the Local Rate of Vertical Diffusion from Dissipation MeasurementsJournal of Physical Oceanography, 10
Compute curvature in C11 function at maximum
D. Brillinger (1985)
FOURIER INFERENCE: SOME METHODS FOR THE ANALYSIS OF ARRAY AND NONGAUSSIAN SERIES DATAJournal of The American Water Resources Association, 21
Loop for power law fit: 19) repeat steps 5-7 using power law instead of Batchelor spectrum; constrain the variance of power law plus noise spectrum to match the observed variance
L. Laurent, R. Schmitt (1999)
The Contribution of Salt Fingers to Vertical Mixing in the North Atlantic Tracer Release ExperimentJournal of Physical Oceanography, 29
Use Eq. (22) to compute standard error in k B
J. Mourn (1996)
Efficiency of mixing in the main thermocline
T. Dillon, D. Caldwell (1980)
The Batchelor spectrum and dissipation in the upper oceanJournal of Geophysical Research, 85
Compute noise spectrum versus frequency (call instrument noise subroutine); change to wavenumbers using drop speed of instrument
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.
Journal of Atmospheric and Oceanic Technology – American Meteorological Society
Published: May 19, 1999
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