Summary and 2018 Update

This website has been dormant since 2015.  The website was started as a platform for discussion related to my book published in 2011.  I have removed some of the blogs, which are out of date.  Blogs that clarified topics in the book, identified errors, and introduced related topics have been retained.

To put this in perspective, the first two book chapters laid the foundation for a complete theory of scintillation, which follows from the forward propagation equation (FPE).  Briefly, for propagation studies Maxwell’s equations are used to derive second-order differential equations that characterize electromagnetic (EM) wave propagation, which are known to engineers as the Helmholtz equation.  However, the Helmholtz equation characterizes EM fields emanating from a collection of interacting point sources.  The problem of interest involves calculating the propagation of an EM field from the phase center of a transmitting antenna to the phase center of a receiving antenna.

To put the Helmholtz equation in a more directly applicable form, it is rewritten as a pair of coupled first-order differential equations that individually characterize EM waves propagating in opposite directions along the reference direction defined by the ray connecting antenna phase centers.  A well-designed radio transmission and reception system will maximize the energy collected at the receiver.  The FPE characterizes the field launched at the transmitter the intercepts the phase center of the receiving antenna.  A further simplification calculates time-harmonic fields, which generally capture all that is needed to calculate signal amplitude, phase, and delay perturbations.  The FPE is exceptionally well suited for calculating propagation effects induces by ionospheric structure intercepted along the propagation path.

As a first-order differential equation, the FPE can be integrated to generate a realization of the propagating EM field.  However, his requires a model of the ionospheric structure.  This information comes from three sources:

  • Purely statistical models that impose an amplitude weighting on independent Fourier modes.
  • Physics-based simulations of ionospheric structure.
  • Configuration space models

Only purely statistical models were introduced in the book.  To accommodate field-aligned anisotropy, the spherical isotropic surfaces of constant spatial coherence are transformed into elongated surfaces, generally ellipsoids.  Additionally, because the transmission paths of primary interest were from earth-orbiting satellites to near-earth receivers, an ionospheric topocentric coordinate system with downward, northward, and eastward reference axes was introduced.   In a downward oriented coordinate system satellite to earth propagation paths intercept the earth’s surface a very large distances from the nadir point.  For efficient computation the integration is performed in a continuously displace coordinate (CDP) system that follows the reference ray.  Chapter 4 in the book developed all the necessary geometric manipulations.  The oblique propagation blog introduces supplemental material to clarifying how CDP coordinates are introduced into moment equations.

While scintillation theory follows from solutions to the FPE, characterizing stochastic structure requires statistical measures.  Statistically homogeneous processes are characterized by spectral density functions (SDFs), which is formally ensemble averages of the intensity of spatial Fourier transformations of the structure realizations.  Chapter 3 reviews the most widely used statistical theory, which introduces a hierarchy of differential equations that statistical observables to functions of the structure SDF.  While much has been learned from solutions to the fourth-order moment equation that characterizes intensity scintillation, the computation involved is too demanding for interpreting diagnostic measurements.

A time-honored simplification is the equivalent phase screen, which concentrates the structure into an equivalent path-integrated phase screen.  Under the narrow-angle scatter or parabolic approximation forward propagation is completely specified by the Fresnel scale.  The scintillation at a fixed distance from the phase screen is reproduced at a different frequency and propagation distance that keeps the Fresnel scale constant. The FPE becomes the more familiar parabolic wave equation (PWE).  Moreover, the scintillation intensity SDF (and scintillation index S4) can be computed a function derived from the structure SDF. The Phase Screen Models for Numerical Computation blog includes a compact summary of the material.

The material in Chapter 4 influenced the development of a simulation model called SIGMA Dispande.   The phase-screen model was used to simulate the GPS effects of scintillation on GPS signals Ghafoori

Chapter 5 addresses system applications, which include signal processing for scintillation diagnostics.  Beacon satellite diagnostic receivers transmit VHF, UHF, and L-band signals.  Digital receives generate frequency-dependent amplitude and phase histories.  The phase data are dominated by changing range, which can introduce Doppler shifts exceeding 12 kHz.  To measure the scintillation component, it is necessary to isolate the Doppler component.  One method is to implement a digital phase locked loop (PLL).  However, the order of the PLL determines the number of samples used to estimate the frequency or phase.  A more robust procedure that allows an arbitrary number of samples is described in the Digital Signal Processing and A Comparison of Phase Locked Loops and Frequency Hypothesis Testing blogs.  The blog AGU 2013 Poster and Supplemental Material describes a true tomographic reconstruction, which is mainly of academic interest because of the sampling requirements.

Chapter 5 also introduced some diagnostic signal processing procedures and a supplemental MatLab library.  Simulated complex scintillation requires no detrending.  For plane-wave excitation the average signal intensity is constant, which is most conveniently set to unity.  Real intensity scintillation measurements vary because of path-loss associated with changing range and antenna pattern variations.  A Butterworth filter is typically used to make a rigid frequency separation between the signal mean and the structure component.  One problem with this approach is that the detrend interval should be data driven and an large as possible.  While it is true that Fresnel filtering suppresses large scale variation, the cutoff is not rigid and level dependent.  We find that wavelet-based filters are easier to implement and more flexible.  Simulated phase scintillation requires no detrending.  Real phase scintillation data is dominated by the geometric path variation.  The problem there is you cannot separate the geometric contribution from total electron content variation, which is the reason navigation satellites use multi-frequency transmissions.  These issues will be discussed in separate blog entries.

What amounts to a GPS Toolbox, which is an upgrade of resources originally developed for low-earth-orbit (LEO) beacon data, contains orbit prediction from ephemeris elements, magnetic field computation, and GPS coordinate manipulation.  Scintillation utilities provide all the geometric transformations needed to accommodate field-aligned geometry dependencies.  A wavelet library provides all the MatLab utilities needed for wavelet based detrending and segmentation applications.

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A Comparison of Phase Locked Loops and Frequency Hypothesis Tracking

Processing low orbiting beacon satellite data has been a challenge largely because of the limited vehicle space for a VHF, UHF, L-band antenna system. The effective antenna patterns severely limit range of uniform illumination range with a common phase center. Low SNR, particularly at acquisition severely, constrains phase-locked-loop (PLL) performance. However, with post-pass processing substantially more processing can be devoted to the critical frequency tracking operation. Several schemes have been used effectively to construct a narrow band filter centered on the Doppler frequency defined over a local time segment. An analysis of these procedures was presented in a paper REF.
In the paper it was hypothesized but not demonstrated that typical PLL performance was degraded for SNRs below 10 dB to the point that direct frequency estimation was a preferable alternative. The note SoftwarePLL compares frequency estimates derived from a conventional PLL and direct frequency estimation.

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IES 2015 Paper

IES2015Rino is an oral presentation of the new material introduced in the previous post Application of Wavelet Based Analysis to C/NOFS Ionospheric Density Measurements.

 

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AGU 2013 Poster and Supplemental Material

Tomographic reconstruction of ionospheric electron density profiles from GPS satellite measurement is being used extensively for global ionospheric monitoring.  For the most part, the reconstruction process is a constrained iterative process.  The results are sensitive to errors and initial conditions.  To study these limitations and the possibility of high-resolution reconstruction two-dimensional simulations have been used.  The simulation and analysis procedures are described in IntermediateScaleTomography, with the accompanying poster RinoPosterSA21B-2019(2).

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Phase Screen Models for Numerical Computation

To the extent that propagation disturbances can be approximated by an equivalent phase-screen, there are analytic models that allow computation of the power-spectrum of the intensity of the field as it propagates away from the phase screen.  The analysis involves an integration to characterize the structure initiation as a phase perturbation and a second integration to propagate the structure to the observation plane.   In real-world applications the integrations are two dimensional.  However, the field-aligned structure in the ionosphere can be exploited to reduce the computations to one-dimensional integrations.

This motivated a revisit of early computations based on two-dimensional and one-dimensional phase screens.  To test the numerical computations, which were performed by Charlie Carrano at the Boston College Institute for Scientific Research, analytic results for unconstrained inverse power-law spectra were used.  Disparities between the computations and the analytic results motivated a careful look at the analytic results derived from complicated limiting operations.  Some errors were found that clarified some long standing disparities between results by Victor Rumsey for isotropic structures and my own two-dimensional results, which should agree with the isotropic results if isotropy was assumed.  The new results are summarized in PowerLawPhaseScreenHighlights, with computational details presented in a separate note. PowerLawPhaseScreenReview,

An updated book errata sheet can be found ScintTheoryErrata

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Beacon Satellite Symposium 2013 Keynote Talk

The following attachment is the keynote talk presented at the Beacon Satellite Symposium, Bath England, July 8, 2013 KeynoteTalk

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Oblique Propagation

Chapter 4 of my book develops a computational framework for beacon satellite propagation.  With a reference coordinate system centered at ionospheric heights, even small propagation angles lead to very large displacements in the observation plane.  To avoid this problem, a continuously displaced coordinate system is used.  The defining equation in Section 4.1 was in error, which combined with some other errors in the defining equations made it difficult to reconstruct the equation upon which most of the calculation in Chapter 4 is based.  Equation 4.5 is correct.  A revision of an earlier derivation is attached.  I’m indebted to Harold Knight and Kshitija Despande for pointing out the inconsistencies. The errata sheet has been updated and placed on the book page.BookNotesNew

The blog entry Statistical Theory of Scintillation Update provides some background material.

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Statistical Theory of Scintillation-Update

In Chapter 3 of my book the system of differential equations that characterize the propagation of complex two-frequency field moments in irregular media were developed. The attached note reviews this material and its connection to the original and most cited source of the equations in the scintillation theory literature.  A number of small but annoying errors in the equations have been corrected, and some definitions have been added for clarity.   With recent updates and additions as of July 23, 2011,  BookNotesCh3fourth-moment-notes, ScintTheoryErrata

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Optical Applications of Full Diffraction

Revised May 30, 2011

Earlier blogs on this topic discussed the relation of  The Theory of Scintillation with Applications to Remote Sensing, which uses the forward propagation method (FPE) , to optical methods based on the Huygens-Fresnel construction.  The revised paper linked below presents a more cohesive development.  There is a regime where wavelength does not dictate sampling requirements.  In this common regime, identical computational methods can be used for optics, acoustics, and radio frequency applications.  This regime includes the Fraunhofer diffraction limit.  However, the FPE is formally unconstrained.

A case in point is ray optics, which is based on a wavelength independent geometrical construction.  An example is shown in the Figure below.  The problem is the omission of diffraction.

The paper uses Fourier domain methods to compute the evolution of the wavefront from the aperture stop where ray optics gives an accurate representation to the focal plane.  The problem is that full diffraction computation requires wavelength scale sampling.

The figure below shows an example of Fraunhofer diffraction where computation at optical wavelengths is easily achieved.

The two figures below show the development of a focus at mm wavelengths where full-diffraction computations are feasible.  The details are discussed in the unpublished paper submitted for publication in Antennas & Propagation Magazine linked below.


Updated June 17, 2011.  OpticsDiffractionFPE2 Continue reading

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What’s the Book About

The theory of scintillation is about electromagnetic wave propagation in structured media.  But why does a phenomenon that has been studied for nearly a century now merit a new theoretical expose?  One part of the answer is that the essential elements of the standard theory are too easily comprehended.
https://chuckrino.com/?p=36
Consider the random lens model mentioned in my earlier post.  That conceptual picture is not far removed from the standard textbook development of the theory,  for example the development of imaging through turbulence in Chapter 8 of Statistical Optics by Joseph Goodman.  The theory provides an explicit representation of wavefront phase, which is very appealing for optical system analyses.  Nonetheless, any theory that builds on a mathematical separation of amplitude and phase cannot be reconciled with the recognized theoretical developments that have evolved over the past 30 years because they address complex field moments.  Conversely, that comprehensive theory (summarized in Chapter 3 of my book) provides very little utility for solving practical problems.

The resolution of the dilemma and the compelling reason for revisiting scintillation theory comes from the computer revolution.  Although the idea is not new, the complete theory of wave propagation in structured media is encapsulated in a comparatively simple first-order differential equation.  The equation admits robust integration using a well-known algorithm that is easily implemented and executed with inexpensive personal computers.  The development is facilitated by interactive languages such as MATLAB.  Thus, simulations based on very high fidelity models bridge the gap between theoretical results that are either over-simplified or unwieldy.   With such simulations one can also explore the combined effects of path-loss, wavefront curvature, and antenna effects.

An example from Chapter 2 of my book (shown in the Figure above) simulates beam propagation along a layer with increasing inward radial refractivity.  As the beam expands, refraction redirects the energy toward the axis where a local focus functions like a local source. 

A second example, which has already been introduced, is strong focusing.  The image shows fine intensity detail that admits no meaningful continuous phase reconstruction.  Yet, back propagation can be used to fully reconstruct the disturbance at the edge of the disturbed region. 

A third example shows the same type of simulation for propagation through highly anisotropic media at oblique incidence.  The overlaid white line and arrow show the apparent scan direction induced by source motion.  The final figure shows the intensity and reconstructed phase along the scan direction, which is what a Beacon  satellite receiver would observe, ideally. 

A directory of MATLAB codes with scripts that will reproduce all the examples in the book and more will be available for download on the MATLAB Central website.

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