The application of numerical simulations in Beacon scintillation analysis and modeling



[1] Modeling Beacon satellite scintillation data presents a number of challenges. The dominant ionospheric structure is anisotropic, and it evolves nonuniformly. Moreover, the length and the orientation of the propagation path that intercepts the structure vary continuously. Thus, even under ideal observing conditions, it is difficult to extract unambiguous driving-point conditions from single-receiver or multireceiver observations. Statistical models are invariably used to interpret scintillation measurements, but the statistical models themselves require a high degree of statistical uniformity that applies only to segments of the data. These challenges are well known, but evolving computer capabilities have provided new opportunities. Modern computer resources support high-fidelity simulations that capture the three-dimensional propagation phenomena in representative propagation environments. Because all aspects of such simulations are known or measurable, one can validate theoretical assumptions and the effectiveness of various analysis procedures. This paper reviews the theory and illustrates the numerical simulation it supports.

1. Introduction

[2] To demonstrate high-fidelity simulations of Beacon satellite data, it is appropriate to start with a signal model. Let v(t) represent a complex time series at the output of a low-noise amplifier (LNA) connected to a receiving antenna. The complex narrowband signal model [Rino, 2011] takes the form

equation image

where η(t) represents unit variance additive Gaussian noise (formally, 〈∣η(t)∣2〉 = 1, with angle brackets denoting an ensemble average) and ϑ(t) represents the effects of propagation disturbances. To isolate the propagation disturbance, the product s(t)ϑ(t) is constructed so that s(t) is the signal that would be received in the absence of channel-induced propagation disturbances (ϑ(t) = 1) and noise. The defining relation is

equation image

The system-dependent factor CF converts signal intensity to signal-to-noise ratio (SNR) units, and w(t) exp {2πifct} represents the Beacon waveform at transmitter frequency fc. The propagation factor exp {2π ik · r(t)}/r(t) is common to the far-field limit of any compact electromagnetic (EM) wave source. The range r(t) is measured from the phase center of the transmit antenna to the phase center of the receiving antenna. The wave vector k has magnitude 2π fc/equation image, where equation image is the mean velocity of light in the background propagation medium. Over sufficiently small time intervals, r(t) equation imager0 + equation image(tt0). Thus, the complex signal model captures the Doppler shift k · equation image/(2π), the signal phase, and the propagation loss factor 1/r(t). (To translate a measured signal to the root SNR form of (2), one first estimates the average noise power. The signal is then scaled by the square root of the average noise power.) Synchronous demodulation of the signal removes the carrier term exp {2πifct}. A filtering operation constrains v(t) to occupy the bandwidth of w(t).

[3] Propagation is characterized by the modified Helmholz equation

equation image

The refractive index n(r) is constrained so that gradients are not steep enough to require explicit treatment [Rino, 2011, chapter 2]. Formally

equation image

where equation image is the uniform background reference level. The consistent definition of k is

equation image

The modified Helmholz equation can be rewritten as

equation image

In the approximation, equation image = 1 and the quadratic term is neglected. To introduce some terminology, consider the solution to (5) when δn(r) = 0. It is well known and easily verified that if ψ(x0, ς) is a solution to (5) in the transverse plane at x = x0, then

equation image

is also a solution to (5) in the displaced transverse planes at x = x0 ± Δx. The solution is given in terms of the two-dimensional Fourier transformation in the initial transverse plane

equation image

[4] In the argument of the first exponential term of (6)

equation image

whereby kg(equation image) is the x component of the wave vector

equation image

The upper sign refers to waves propagating along the positive or forward x direction. The lower sign refers to waves propagating along the negative or backward x direction. With the Taylor series expansion of g(equation image) in mind, let ikΘψ± represent the linear operation defined by (6). Formally,

equation image

where ∇ is the transverse Laplacian operator. The more familiar parabolic approximation truncates the Taylor series expansion of (6) or (9) at the first nonzero term. However, for the split-step numerical computation described below, this narrow-angle scatter approximation offers not advantage.

[5] Following the development in the work by Rino [2011, chapter 2], it can be shown that the following pair of coupled first-order differential equations is equivalent to (5):

equation image


equation image

represents the total wavefield at x. The coupled propagation equations account for the fact that the interaction of the total wavefield with refractive index structure acts as a local source of waves propagating both in the forward direction and the backward direction. However, if the external source launches waves predominately in the forward direction, the backscatter (backward propagating) contribution is very small. Without dismissing backscatter entirely, it is a very good approximation to replace ψ(x, ς) in the forward (10) with ψ+(x, ς). This leads to the forward propagation equation (FPE)

equation image

The FPE, which encompasses refraction, diffraction, and the effects of small-scale structure, is the most general starting point for scintillation analysis.

[6] The FPE is readily solved numerically by using the split-step method. The propagation medium is divided into layers over which the media interaction contribution is replaced by an integrated phase perturbation. The diffraction operator (6) is then applied as if the segment were uniform. Both the split-step scheme and alternative finite difference methods involves some degree of approximation; however, ease of implementation, insight, and robustness generally favor the split-step method. The split-step layer thickness is dictated by the requirement that amplitude change induced by the diffraction operator be small. Transverse sampling is dictated by the spatial wave number content of the field, but there is no explicit requirement that the range of scattering angles be small. If one is concerned about the forward approximation itself, the FPE solution can be used as a source field that initiates cumulative backscatter via the solution to the backward (10) equation. The cumulative backscatter intensity must remain a small fraction of the forward intensity. The phase-screen model is formally identical to the split-step solution to the FPE, but there is no requirement that the phase structure be uncorrelated from screen to screen.

[7] To complete this introduction, the relation between ϑ(t) in (1) and solutions to the FPE are established. The formal definition of ϑ(t) normalizes the FPE solution to the far-field approximation, which can be established analytically once the aperture distribution is specified. However, experience has shown that using plane wave excitation, which propagates without amplitude change in a uniform propagation medium, provides a viable alternative that greatly reduces the computational demands. Applying a well-known spherical wave correction to the plane wave solution generates ϑ(t). In summary, the FPE is effectively a model that can be employed to evaluate all aspects of scintillation observations. What remains is to incorporate background models, structure models, and Beacon satellite propagation scenarios.

2. The Statistical Theory of Scintillation

[8] Scintillation theory, as it has evolved over the past four decades, often makes no distinction between the theory of scintillation as reviewed in the introduction and the statistical theory of scintillation, which assigns a statistical model to the refractive index fluctuations δn(x, ς). Scintillation theory is driven by path integrals over the structure, which are used directly in the split-step solution to the FPE. Consider the simplest form of the path integral

equation image

It is straightforward to show that if δn(x, ς) decorrelates over all x displacements within ∣x∣ < L/2, then in expectation

equation image

where Φδϕ(K) is the two-dimensional spectral density function (SDF) of the phase path integral and Φδn(Kx, K) is the three-dimensional SDF of the in situ refractive index structure. This proportionality greatly simplifies theoretical computations. Thus, the requirement of decorrelated phase structure within a layer is imposed to provide analytically tractable results.

[9] The essential elements of the statistical theory are well known. For example, in a measurement plane normal to the propagation path traversing a disturbed region, the phase and intensity SDFs under weak-scatter conditions are given by the Fresnel-filtering relations

equation image

where x is the distance from the centroid of the disturbed region. The Fresnel wave number, equation image, establishes a spatial frequency below which the structure SDF contribution to intensity is strongly suppressed. Thus, near the disturbance the intensity scintillation is small, whereas phase scintillation responds to the largest structure scales that can be measured. The scintillation index,

equation image

where δI = I/〈 I 〉, is formally the square root of the integral of ΦI(κ) over κ. From the weak-scatter theory one would expect a monotonic increase of SI with increasing x. The strong-scatter theories predicts convergence of SI to unity at sufficiently large distances from the disturbed region. However, under appropriate conditions, SI can achieve values much larger than unity at intermediate distances.

[10] To illustrate this behavior, an isotropic power law SDF, Φδn ∝ ∣K−(2ν+ 1), with 2ν + 1 = 5.8 used to generate a single phase screen structure realization. The 3-D power law index is steeper than the Kolmogorov index 11/3, which does not produce strong focusing. In Kolmogorov turbulence, SI increases monotonically to one. Figure 1 (top) shows the scintillation index derived from the simulated wavefield at 600 km. The SI maximum at x = 540 m is attributed to enhanced large-scale structure forming a strong lens-like focus. In the early scintillation literature this phenomenon has been referred to as strong focusing. As a measure of the departure from the expected exponential statistics limit, Figure 1 (bottom) shows the measured fractional moments, FM(m), which are defined as

equation image

plotted against FM(2). The pentagrams mark the fractional moments for exponential distribution form FM(m) = m!. In the strong focusing regime where SI > 1, the departure from exponential (Rayleigh) statistics is extreme.

Figure 1.

(top) Measured SI and (bottom) fractional moments for realizations of scintillation from a power law phase screen with ν = 2.4 realization.

[11] This is reinforced in Figure 2, which shows the intensity structure at the SI maximum. Although this extreme behavior would not be expected for the conditions that are known to cause ionospheric scintillation, local enhancements due to scale sampling of large enhancements can create local intensity structure enhancements that are distinctly nonhomogeneous.

Figure 2.

Intensity structure at the distance from the phase screen where SI achieves its maximum value.

3. Beacon Satellite Scintillation

[12] Structured regions that produce the scintillation observed on Beacon satellite transmissions are highly nonuniform. Thus, the statistical homogeneity that supports a spectral characterization applies only over local segments within a more variable background. Segmentation procedures have been developed to identify and characterize homogeneous subregions [Mallat et al., 1998], but the scintillation problem is compounded by the complex relation between field observables and δn(x, ς). Insofar as the FPE solution is concerned, no distinction need be made between structure and background, but a parametric structure model remains essential to modeling. The Beacon satellite modeling challenges, as summarized by Bhattacharyya et al. [1992], are well known.

[13] Figure 3 shows a local reference coordinate system (xpypzp), which has been used extensively for scintillation analysis. A ray from a source through a structured region to a measurement plane involves a very large volume of the propagation space. As discussed by Rino [2011, chapter 4], an extension of the theory by Costa and Basu [2002] allows efficient simulation and theoretical computation by introducing a continuously displaced local coordinate system that follows the central ray shown in Figure 3. FPE-based numerical computation in the displaced coordinates are implemented by the spatial variable change

equation image

and the replacement

equation image

in the Fourier domain form of the diffraction operator. The unit vector equation imageequation image is the renormalized projection of the propagation vector onto the transverse plane. Within the constraints that support (14), results for isotropic normal incidence can be translated algebraically. The details are summarized with examples in the work by Rino [2011, chapter 4], and will not be repeated here. Managing the continuous displacement, however, introduces some modeling complications discussed below.

Figure 3.

Reference geometry for Beacon satellite to ground propagation.

3.1. Equatorial Beacon Satellite Geometry

[14] To establish a geometry and a consistent set of associated parameters for simulation, the C/NOFS satellite pass described by Rino [2011, chapter 4] is used. Figure 4 shows the trajectory of the 400 km penetration point (dark line) overlaid on a map of the magnetic dip angle. (Zero dip angle defines the geomagnetic magnetic equator.) The pentagram marks the location of the receiving station. The pass cuts field lines, which are perpendicular to the contours in Figure 4, at a small oblique angle. The pass geometry establishes the pointing angles in the reference coordinate system, the orientation of the magnetic field, and the apparent drift rate in the measurement plane induced by the satellite motion. The satellite-induced apparent drift rate is augmented by true drift motion of the irregularities.

Figure 4.

Trajectory of 400 km penetration point overlaid on B field dip angle map.

3.2. Equatorial Structure Model

[15] Equatorial scintillation has long been associated with very large scale structures called plumes. In a purely statistical model in which each slab is brought into a split-step computation as an independent realization, the structure variation must be imposed analytically. Costa used simultaneous radar backscatter measurements to impose such a spatial variation. His results are described in this special Radio Science issue. Here numerical simulations that incorporate the known physical processes that generate equatorial plumes are used [Retterer, 2010a].

[16] The simulations generate three-dimensional maps of the background electron density. However, because of the very large computation grid, the smallest resolved scale is several kilometers. Moreover, numerical diffusion distorts the structure at the smallest resolved scales. The small-scale structure that drives scintillation must be imposed. Retterer [2010b] used known spectral characteristics of the scintillation in the small-scale regime to extrapolate the resolved simulation structure down to the scale sizes responsible for scintillation. Retterer's [2010b] scintillation modeling approach used a path-integral scheme first introduced by Costa and Kelley [1977]. The computational efficiency of the path-integral approach allowed Retterer to produce a time history of the expected scintillation associated with an entire plume cycle. Retterer has kindly made his simulation results available for this preliminary study.

[17] Because of the very large amount of data that comprises a complete plume cycle, only a snapshot of the plume evolution at its peak is analyzed here. The intent is to generate preliminary results that will stimulate further research. Figure 5 shows the imposed zonal plume structure in a topocentric coordinate system centered on the ground receiving station. Each of the seven rays shown in white is the east–west projection of a path from the satellite to the ground station. The east–west reference plane and the plume zonal plane do not coincide, but for demonstration purposes this distortion has been ignored. The satellite positions were selected to simulate scintillation snapshots of a plume traversal. The horizontal lines show slab plane boundaries for split-step integration along the path. The results presented below will be referred to by number from left to right.

Figure 5.

Simulated plume structure with real satellite propagation paths superimposed. The plume simulation has been interpolated, shifted, and converted to refractivity units. The horizontal lines are projections of the slabs used in the FPE solution.

[18] Electron density is converted to refractive index by using the relation

equation image

where re = 2.8197402894 × 10−15 m is the classical electron radius, c is the velocity of light in the background medium, and f is the frequency. Because n in a dispersive plasma is less than unity, the refractivity units shown are defined as (n − 1)106.

[19] To generate the phase perturbation induced by each slab it is necessary to integrate the 3-D structure within the slab. However, because of the very different mesh sizes for propagation and plume simulation, resampling the coarse simulations must be used. Any such interpolation operation will populate the small-scale spectral regime that supports the scintillation. The interpolated data were filtered to remove any spurious small-scale structure in the scintillation-scale range. Additionally, excising the background to populate the truncated slabs (see Figure 5), introduces edge effects that are controlled by field tapering.

[20] The complete refractive index model is defined by the relation

equation image

where nt(r) is the background component just described. Structure component realizations, δn(r), are generated from model SDFs. The simplest defining relation is

equation image

where Cs is the turbulent strength. To the extent that the Kx = 0 approximation is valid, the model realization consistent with the geometry take the form

equation image

where the coefficients A, B, and C depend on the propagation angles and the anisotropy of the structure. However, power law behavior cannot persist over all scale sizes. The small-scale cutoff imposed by diffusion has little effect on scintillation. At the other extreme, large-scale structure persists to the defining profile dimensions. For the simulations presented here, the following two-component model is used:

equation image


equation image

If ν1 = −1/2, qL functions as a conventional outer-scale cutoff. Taking ν1 and ν2 to be positive with ν2 > ν1 more closely represent in situ structure measurements. A value of qL corresponding to 600 m is typically reported [Kelley, 2008, chapter 4]. See also Figure 5 in the review paper by Hysell [2000], which is particularly useful for visualizing the 3-D structure environment.

3.3. Preliminary Results

[21] The output of an FPE simulation is a static complex field that accommodates each displaced slab contribution. However, the free space propagation segment that generates the field in the measurement plane centered on the receiving antenna phase center can be computed in one step. Movement of the raypath through the structured region creates the time series at the ground observation point that is actually measured. To construct a realization of this time series, the translational velocity at the origin of the 400 km reference coordinate system is used to define a constant one-dimensional measurement-plane trajectory. To the extent that the geometry is invariant, the mean motion of the propagation path is equivalent to scanning measurement plane along the measurement-plane trajectory. Figure 6 shows a 400 MHz signal intensity realization of the Path 1 measurement plane with the scan line superimposed. The small skew of the striations is a consequence of the angle of the plane of the pass from the magnetic field direction (see Figure 4).

Figure 6.

Path 1 field intensity in a plane centered on the receiving antenna. The white line lies along the scan direction of the reference coordinate system at 400 km.

[22] To show the effects of background structure, the simulations were performed for the background plume without structure and with structure. Figure 78 shows a summary of the 400 MHz raw intensity (Figure 7, top) and reconstructed phase (Figure 7, bottom). A 512 point centered boxcar average is superimposed (blue). One can see immediately that the intensity variation imposed by the background plume structure is small. Paths 3 and 4 show some ringing structure, most likely from an edge gradient. Direct interception of the plume does induce a small focusing effect. The focusing is more pronounced if the propagation path is aligned with the plume wall. At lower frequencies, the induced intensity variations would be much larger. However, as one should expect, the plume structure totally dominates the phase. This is seen most prominently in Pass 5 between 100 and 120 s. Figure 8 shows the same summary for the simulations with structure imposed. Here the structure completely dominates the intensity variation while the corresponding phase structure, although large by scintillation measures, is small. This is an encouraging result for total electron content (TEC) analyses, particularly tomographic reconstruction, where scintillation is a nuisance that complicates the TEC interpretation.

Figure 7.

Summary of 400 MHz (top) intensity and (bottom) phase for seven paths chosen to summarize an unstructured plume transversal.

Figure 8.

Summary of 400 MHz (top) intensity and (bottom) phase for seven paths chosen to summarize a structured plume transversal.

[23] The full ramifications of structure intermingling for data analysis remain to be established. Standard analysis procedures were applied to the Path 1 data. The results are summarized in Figures 9 and 10. Figure 9 (top) shows the intensity record in more detail. The scintillation index (SI) shown in Figure 9 (bottom) was computed by using the boxcar average as an estimate of the local mean. The intensity normalized to a local estimate of the mean is usually referred to as detrending. The detrended result depends on the window, and a proper window choice would use segmentation procedures alluded to earlier. In any case, the surviving SI variation should be thought of as stochastic fluctuation induced by large-scale power law structure. The predicted weak scatter SI for the imposed structure is larger than unity, but the weak scatter theory necessarily over predicts SI under strong scatter conditions. Figure 10 shows the intensity and phase spectra derived from the entire record. The reason for using the entire record rather than applying standard segment-and-average schemes is to preserve the maximum possible large-scale resolution. For the same reason, no windowing is used. The theory curve in Figure 10 is computed for the integrated phase of a single-power law with the high-frequency index. There are numerous factors to contend with in making such a computation, but an important simulation objective is to learn how to quantitatively extract the known in situ spectral model parameters from realistic model computations.

Figure 9.

Detrended (top) intensity and (bottom) scintillation index for Path 1 with structure.

Figure 10.

Intensity (gray) and phase (black) SDF measured from Path 1 plume with structure. Theory (dashed line) is derived from integrated phase model.

[24] Figures 11 and 12 show the same results for the Path 5 simulations, which intercept the most highly structured region. Overall, the SI index is reduced, which is attributed to the low-density regions that contribute little to the cumulative propagation. There is little difference in the overall spectral characteristics, other than the failure of the scaling procedure to align the theory with data. This result is encouraging because it suggests an intrinsic separation by scale when conventional analysis procedures are used.

Figure 11.

Detrended (top) intensity and (bottom) scintillation index for Path 5 with structure.

Figure 12.

Intensity (gray) and phase (black) SDF measured from Path 5 plume with structure. Theory (dashed) is derived from integrated phase model.

4. Conclusions and Future Work

[25] The main purpose of this paper was to demonstrate fully three-dimensional simulations of Beacon satellite propagation through a turbulent structure embedded in a representative large-scale background. Similar results have been presented by Bernhardt et al. [2006]. The results are similar, but Bernhardt et al. used an equivalent phase screen model for both the background and structure. However, Bernhardt's [2007a, 2007b] quasi-analytic plume model is more efficient than a full numerical simulation. The encouraging result here is that it is quite feasible to explore the ramifications of propagation disturbances in realistic backgrounds. The increased computation required to solve the FPE using the structure to dictate the sampling requirements is not prohibitive. The challenge is to define a refractive index model that fully accommodates the plume background and the structure. In all the modeling approaches structure is imposed with some degree of ad hoc scaling. The model could be refined with physics-based procedures that, for example, used the coarsely sampled plume background as an initial condition for a local structure simulation that generates small-scale structure. One expects stronger turbulence as the path approaches the plume depletion [Basu et al., 1983; Whalen, 2009]. Considerably more work needs to be done to explore the use of multifrequency data and multistation data for tomographic reconstruction. These procedures that emerge should allow estimation of parameters that validate physics-based models of both background and structure characteristics.

[26] The 3-D propagation code that was used for these computations can be downloaded with examples from the MATLAB Central File Exchange (