## 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

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

The system-dependent factor *CF* converts signal intensity to signal-to-noise ratio (SNR) units, and *w*(*t*) exp {2*πif _{c}t*} represents the Beacon waveform at transmitter frequency

*f*

_{c}. The propagation factor exp {2

*π i*

**k**·

**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 2

*π f*

_{c}/, where is the mean velocity of light in the background propagation medium. Over sufficiently small time intervals,

**r**(

*t*)

**r**

_{0}+ (

*t*−

*t*

_{0}). Thus, the complex signal model captures the Doppler shift

**k**· /(2

*π*), 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 {2

*πif*}. A filtering operation constrains

_{c}t*v*(

*t*) to occupy the bandwidth of

*w*(

*t*).

[3] Propagation is characterized by the modified Helmholz equation

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

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

The modified Helmholz equation can be rewritten as

In the approximation, = 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*(*x*_{0}, ς) is a solution to (5) in the transverse plane at *x* = *x*_{0}, then

is also a solution to (5) in the displaced transverse planes at *x* = *x*_{0} ± Δ*x*. The solution is given in terms of the two-dimensional Fourier transformation in the initial transverse plane

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

whereby *kg*() is the *x* component of the wave vector

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*() in mind, let *ik*Θ*ψ*^{±} represent the linear operation defined by (6). Formally,

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):

where

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)

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.