Power-Law Parameter Estimation

The basic structure and dynamics of the ionosphere are captured by deterministic models.  At the next level, physics-based ionospheric simulations produce stochastic structure, which is characterized by a statistical model.  Statistical models characterize an ensemble of realizations that support well-defined average measures, such as means, higher moments, probability density functions, and covariance.  Such models require statistical uniformity, whereas real processes not strictly homogeneous.

Analyzing real diagnostic data requires segmentation and averaging.  The length of the segment and the sample interval determine the scale or frequency measurement range.  Selecting a segmentation requires and assessment of the largest scale that supports statistical substructure.  In our 2014 paper Wavelet-based analysis and power law classification of C/NOFS high-resolution electron density data, REF wavelets were introduced because of their ability to measure the variation of the scale over as many scale replications supported by the interval.  Scale spectra measure scale versus distance, and thereby provide scale measures and scale dependent uniformity.  A classification procedure was then developed to estimate two-component power-law parameters.

In a second study the classification procedure was applied to C/NOFS data accumulated over a four-year period REF.  It was found that the most highly disturbed passes supported two-component power-law, giving way to a single power-law at smaller disturbance levels.  All the classified SDFs showed a correlation between the turbulent strength and the large-scale spectral index.  It had also been noted that the correlation was observed in remote diagnostics as well.

Because of the persistence of the correlation between the turbulent strength and the large-scale power-law index and some published papers discovered in development of Maximum Likelihood IPE, we began to suspect that the correlation was intrinsic to power-law parameter estimation.  A detailed study, which has been submitted for publication, REF, show that this was indeed the case.

It was shown further that wavelet-based estimators exaggerate the correlation.  The ramifications for the C/NOFS results change mainly the interpretation.  The correlation is intrinsic power-law measurements, not a property of the underlying structure.  Regarding wavelets, we believe they are useful for identifying segmentation, but should not be used in place of periodograms for IPE.

About Chuck

Retired research engineer. Recently published book "The Theory of Scintillation with Applications in Remote Sensing," John Wiley IEEE Press, 201
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