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).
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
The following attachment is the keynote talk presented at the Beacon Satellite Symposium, Bath England, July 8, 2013 KeynoteTalk
The following attachment was presented at the CEDAR workshop Friday 29 June at Boulder Colorado. CEDAR-Talk-Rino
The following attachment was presented at the International Center for Theoretical Physics, Tereste Italy, on May 14, 2013 GNSS_Rino
Physics based models can generate time-dependent three-dimensional realizations of ionospheric Equatorial Spread F (ESF). Although these very large-scale models have limited resolution (~5 km), they provide extremely useful tests beds for exploring three-dimensional anisotropic structure. The attached note describes an application to interpreting in-situ scans of the structure. PhysicsBasedModels
Refractive index models can be thought of as the interface between measurement phenomenology and the physics of the atmosphere and ionosphere. The refractive index determines how electromagnetic was will interact with the structure. A plethora of physical process act to structure the particle density profiles that in turn determine the refractive index. The attached paper reviews and extends the structure models used for propagation analyses is disturbed media. StructureModelsDataInterpretation
The examples used to illustrate the statistical theory of scintillation as developed in Chapter 3 of The Theory of Scintillation with Applications in Remote Sensing are all based on a three-dimensional statistical model. The three-dimensional model assumes statistical homogeneity in the three cardinal directions, but not isotropy. Surfaces of constant correlation are closed, but need not be spherical. Anisotropy is important for radio propagation in the earth’s ionosphere because the charged particles that comprise the F-region plasma move more freely along magnetic field lines than across them. The field-aligned distances over which structures are essentially invariant can be so large that the structure is more accurately modeled with a two-dimensional stochastic model.
The report StructureModelsDataInterpretation reformulates the structure model examples to include two-dimensional structures. The end result is a two-dimensional propagation code in which only the random cross-field structure influences the propagation. The two-dimensional model provides a physics-based rationale for the common practice of using two-dimensional propagation codes for simulation.
With regard to measurements, both in-situ probes and propagation measurements provide one-dimensional time series that must be analyzed to extract the model parameters that characterize the structure. The one-dimensional scans can be fit to either two- or three-dimensional structure models. To determine which model best explains data patterns requires careful analysis. The model results can guide such efforts.
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.
Fractional Brownian Motion (fBm) is a model for nonstationary (inhomogeneous) processes developed by Benoit Mandlebrot. It shares a self-similar scaling property that evolved from the study of fractals as a model for real physical structures. A realization one-dimensional Browning motion can be generated by integrating white noise. Although Brownian motion is non-stationary, it does support stationary increments and an inverse frequency squared power law. The fBm process generalizes this property to multiple dimensions and a range of power-law structures. The trend-like departures from homogeneity associated with equatorial spread F suggest that fBm might be a good model for ionospheric irregularities. This possibility has been explored following our last three blogs.
The figure below shows an example of two-dimensional fBm generated by Fourier filtering:
The analysis and details can be found in the paper FractionalBrownianMotion .