Determination and Sensitivity Analysis of the Specular Reflection Point in GNSS Reflectometry

In applying Global Navigation Satellite System Reflectometry (GNSS-R) techniques for the remote sensing of surface properties of the Earth, it is imperative to determine the specular reflection point and provide a quantitative characterization of the associated errors. In this article, a rigorous formulation of the problem with respect to ellipsoidal Earth is provided for the determination and error analysis of the specular reflection point. A polynomial equation approach is developed to characterize the specular reflection point. This explicit characterization is beneficial in the GNSS-R receiver operation. The sensitivity analysis is further performed to assess the errors in the presence of uncertainties.

SECTION I. Introduction

The Global Navigation Satellite System Reflectometry (GNSS-R) technique has been regarded as an enabling technique in revealing geophysical parameters from reflecting areas of the Earth, which can potentially be used in altimetry, ocean wind retrieval, and soil moisture measurement [1], [2], [3]. Through a multistatic setup, a GNSS-R mission at low Earth orbit (LEO) can obtain a mesoscale measurement of the Earth's surface by processing the scattered/reflected GNSS signals. As a result, several space GNSS-R missions have been conducted or planned, as tabulated in Table I.

TABLE I Spaceborne GNSS-R Missions

The received Global Navigation Satellite System (GNSS) signals are known to be weak, and the reflected signals are even weaker. To receive and process the reflected/scattered GNSS signals, a long integration through a combination of coherent and noncoherent processing is needed to enhance the signal quality. Since the main lobe of the reflected/scattered signals is centered around the path from the transmitter to the specular reflection point and then to the receiver, it is imperative to determine the specular reflection accurately. The GNSS-R receiver demands this information to assess the code phase and its variation in the generation of local replica for correlation operations so that the scattered signals can be received and the delay Doppler maps (DDMs) can be constructed. In forming the DDMs, the computation of the specular reflection point is a routine task in a GNSS-R receiver. For example, in the TRITON mission [4], eight reflection points are computed in every second. It is, thus, desired to perform this computation effectively. In addition to the determination of the specular reflection point, the sensitivity needs to be addressed. Typically, the specular reflection point is computed based on the position of the GNSS satellite, the position of the receiver, and the reference model of the Earth. Nowadays, the positions of the GNSS satellite and the receiver are available by processing direct line-of-sight signals. It is, thus, of interest to investigate the sensitivity analysis in the determination process.

The subject about the determination of the reflection point has been investigated in the past. In [12], the spherical Earth is assumed, and it is shown that by solving a quartic (fourth-order) polynomial equation, the specular reflection can be computed. As the Earth is often modeled as an ellipsoid, an analytic treatment of the specular reflection point on the ellipsoidal Earth has not been discussed in the past, to our knowledge. One can use an iterative method to deal with the ellipsoidal Earth, as discussed in [13] and [14]. In [15], a quasi-spherical approximation approach is developed to account for the problem. In this article, the properties of the specular reflection point on the ellipsoidal Earth are examined. A polynomial approach is developed to characterize the specular reflection point. Such an analytic characterization is beneficial in two regards. The approach is not subject to uncertainty in the initial estimate, and the complexity in iterations is avoided. To address the sensitivity issue with respect to perturbations on the positions of the transmitter and the receiver, this article proposes the use of the concept of the dilution of precision (DOP) [16], [17], which is a well-known metric in relating the measurement error to the positioning error in GNSS literature, to account for the errors. Two DOP metrics in reflection are proposed in relating the transmitter/receiver error to the position error of the specular reflection point. With these metrics, one can then assess the error ellipses on the surface as the GNSS-R receiver orbits around the Earth.

A preliminary version of this article is published in [18]; this article provides a more succinct characterization of the reflection point. Moreover, sensitivity analyses and simulation examples are provided to illustrate the approach. The rest of this article is organized as follows. In Section II, three properties about the specular reflection are discussed and utilized to establish an algebraic characterization. In Section III, the sensitivity analysis of the computed reflection point with respect to variations on the positions of the transmitter and the receiver is then conducted. Metrics of DOP in reflection are proposed to assess the accuracy in the determination process. Simulation analysis results are presented in Section IV to illustrate the proposed approach and accuracy analysis. Finally, Section V concludes this article.

SECTION II. Specular Reflection Point: Properties and Its Determination

In this section, the properties of the specular reflection point are exhibited and, through a combination of these properties, a polynomial characterization is established and a computationally feasible approach is developed. In GNSS-R operation, it is known that the specular reflection is required to satisfy the following properties.

1. The specular reflection point is on the surface of the Earth.

2. The minimal distance property is satisfied, that is, the distance of the propagation path from the transmitter to the specular reflection point and then to the receiver is minimal among all admissible paths.

3. The incidence angle at the specular reflection point is the same as the reflection angle.

These three properties will be explored to render a characterization of the specular reflection point.

Let $\mathbf {p}_{s} \in \mathbb {R}^{3}$ be the specular reflection point in the Earth-centered Earth-fixed (ECEF) coordinate system. An ellipsoidal Earth is considered in the following. The first property, thus, implies that the point $\mathbf {p}_{s}$ is constrained by the following quadratic equation: $$\begin{equation*} \mathbf {p}_{s}^{T} \mathbf {E}^{-1} \mathbf {p}_{s} = 1 \tag{1} \end{equation*}$$ where $\mathbf {E}= \left[ \scriptsize\begin{array}{ccc}a^{2} & 0 & 0 \\ 0 & a^{2} & 0 \\ 0 & 0 & b^{2} \end{array} \right] = \text{diag} \lbrace a^{2}, a^{2}, b^{2} \rbrace$ and the variables $a$ and $b$ are the semimajor axis and semiminor axis of the ellipsoidal Earth, respectively. In the above, the superscript $(\cdot)^{T}$ represents the transpose operation, while $(\cdot)^{-1}$ stands for the inverse of the matrix. In the World Geodetic System 1984, the semimajor and semiminor axes are $a = \text{6 378 137}$ m and $b = \text{6 356 752.3142}$ m, respectively.

Let $\mathbf {p}_{t} \in \mathbb {R}^{3}$ and $\mathbf {p}_{r} \in \mathbb {R}^{3}$ be the positions of the transmitter (GNSS satellite) and the receiver (LEO satellite), respectively. The distance of the overall propagation path from the transmitter to the specular reflection point and then to the receiver is $$\begin{equation*} d = \Vert \mathbf {p}_{t} - \mathbf {p}_{s} \Vert + \Vert \mathbf {p}_{s} - \mathbf {p}_{r} \Vert \tag{2} \end{equation*}$$ where $\Vert \cdot \Vert$ stands for the norm of the vector. The minimal distance property characterizes the specular reflection point as $$\begin{equation*} \mathbf {p}_{s} = \arg \min _{\mathbf {p}, \, \mathbf {p}^{T} \mathbf {M}^{-1} \mathbf {p}= 1} \left(\Vert \mathbf {p}_{t} - \mathbf {p}\Vert + \Vert \mathbf {p}- \mathbf {p}_{r} \Vert \right). \tag{3} \end{equation*}$$ The third property is essentially the Snell's law, which implies that $\mathbf {p}_{s}$ satisfies $$\begin{equation*} \frac{\mathbf {w}^{T} (\mathbf {p}_{s} - \mathbf {p}_{t}) }{\Vert \mathbf {p}_{s} - \mathbf {p}_{t} \Vert } = \frac{\mathbf {w}^{T} (\mathbf {p}_{s} - \mathbf {p}_{r}) }{\Vert \mathbf {p}_{s} - \mathbf {p}_{r} \Vert } \tag{4} \end{equation*}$$ where $\mathbf {w}$ is the normal vector with respect to the local tangential plane of the ellipsoidal Earth at $\mathbf {p}_{s}$. The normal vector in (4) at $\mathbf {p}_{s}$ is given by $\mathbf {w}= 2 \, \mathbf {E}^{-1} \, \mathbf {p}_{s}$ [19]. Thus $$\begin{equation*} \frac{\mathbf {p}_{s}^{T} \mathbf {E}^{-1} (\mathbf {p}_{s} - \mathbf {p}_{t}) }{\Vert \mathbf {p}_{s} - \mathbf {p}_{t} \Vert } = \frac{\mathbf {p}_{s}^{T} \mathbf {E}^{-1} (\mathbf {p}_{s} - \mathbf {p}_{r}) }{\Vert \mathbf {p}_{s} - \mathbf {p}_{r} \Vert } \tag{5} \end{equation*}$$ which, through the substitution of (1), can be expressed as $$\begin{equation*} \frac{1 - \mathbf {p}_{s}^{T} \mathbf {E}^{-1} \mathbf {p}_{t} }{\Vert \mathbf {p}_{s} - \mathbf {p}_{t} \Vert } = \frac{1 - \mathbf {p}_{s}^{T} \mathbf {E}^{-1} \mathbf {p}_{r} }{\Vert \mathbf {p}_{s} - \mathbf {p}_{r} \Vert }. \tag{6} \end{equation*}$$

A. Orthonormal Coordinates

In analyzing the reflection geometry, it is convenient to transform the coordinates into a canonical form. A procedure is outlined in this subsection. Assume that $\mathbf {p}_{t}$ and $\mathbf {p}_{r}$ are linearly independent. Then, one can employ the Gram–Schmidt process [20] to render an orthonormal coordinates system in which the first axis is given by the normalization of the vector $\mathbf {p}_{t}$: $$\begin{equation*} \mathbf {n}_{1} = \frac{\mathbf {p}_{t}}{\Vert \mathbf {p}_{t} \Vert }. \tag{7} \end{equation*}$$ The second axis is obtained from the Gram–Schmidt orthonormalization, as follows: $$\begin{equation*} \mathbf {n}_{2} = \frac{\mathbf {p}_{r}-(\mathbf {p}_{r}^{T}\mathbf {n}_{1})\mathbf {n}_{1}}{\Vert \mathbf {p}_{r}-(\mathbf {p}_{r}^{T}\mathbf {n}_{1})\mathbf {n}_{1} \Vert }. \tag{8} \end{equation*}$$ In addition, the third axis $\mathbf {n}_{3} = \mathbf {n}_{1} \times \mathbf {n}_{2}$, which is the cross product of the first two axes, is defined to constitute a right-handed coordinates system. As $\lbrace \mathbf {n}_{1}, \mathbf {n}_{2}, \mathbf {n}_{3} \rbrace$ is an orthonormal basis, the specular reflection point can be uniquely parameterized as $$\begin{equation*} \mathbf {p}_{s} = a \left(s_{1} \, \mathbf {n}_{1} + s_{2} \, \mathbf {n}_{2} + s_{3} \, \mathbf {n}_{3} \right) = a \, \mathbf {N}\, \mathbf {s} \tag{9} \end{equation*}$$ for some $s_{1}$, $s_{2}$, and $s_{3}$. The matrix $\mathbf {N}$ and vector $\mathbf {s}$ in (9) are given by $\mathbf {N}= [ \begin{array}{ccc}\mathbf {n}_{1} & \mathbf {n}_{2} & \mathbf {n}_{3} \end{array} ]$ and $\mathbf {s}= a^{-1} \, \mathbf {N}^{T} \, \mathbf {p}_{s} = [ \begin{array}{ccc}s_{1} & s_{2} & s_{3} \end{array} ]^{T}$, respectively. The vector $\mathbf {s}$ can, thus, be regarded as the normalized position of the reflection point in the $\lbrace \mathbf {n}_{1}, \mathbf {n}_{2}, \mathbf {n}_{3} \rbrace$ coordinates. Let $t_{1} = \Vert \mathbf {p}_{t} \Vert /a$, $r_{1} = (\mathbf {p}_{r}^{T} \mathbf {n}_{1})/a$, and $r_{2} = \Vert \mathbf {p}_{r} - (\mathbf {p}_{r}^{T} \mathbf {n}_{1}) \mathbf {n}_{1} \Vert /a$; then, the normalized positions of the transmitter and receiver in the $\lbrace \mathbf {n}_{1}, \mathbf {n}_{2}, \mathbf {n}_{3} \rbrace$ coordinates are, respectively, given by $\mathbf {t}= a^{-1} \, \mathbf {N}^{T} \, \mathbf {p}_{t} = [ \begin{array}{ccc}t_{1} & 0 & 0 \end{array} ]^{T}$ and $\mathbf {r}= a^{-1} \, \mathbf {N}^{T} \, \mathbf {p}_{r} = [ \begin{array}{ccc}r_{1} &r_{2} & 0 \end{array} ]^{T}$. Note that the third components of $\mathbf {t}$ and $\mathbf {r}$ are zeros.

Let $\mathbf {S}= a^{-2} \, \mathbf {N}^{T} \, \mathbf {E}\, \mathbf {N}$. Then, from (1) and (9), the specular reflection point $\mathbf {s}$ satisfies the following constraint: $$\begin{equation*} \mathbf {s}^{T} \, \mathbf {S}^{-1} \, \mathbf {s}= 1. \tag{10} \end{equation*}$$ In addition, under the coordinates $\lbrace \mathbf {n}_{1}, \mathbf {n}_{2}, \mathbf {n}_{3} \rbrace$, the relationship (6) becomes $$\begin{equation*} \frac{1 - \mathbf {s}^{T} \, \mathbf {S}^{-1} \, \mathbf {t}}{\Vert \mathbf {s}- \mathbf {t}\Vert } = \frac{1 - \mathbf {s}^{T} \, \mathbf {S}^{-1} \mathbf {r}}{\Vert \mathbf {s}- \mathbf {r}\Vert }. \tag{11} \end{equation*}$$ Furthermore, since $\mathbf {N}$ is an orthogonal matrix, the minimal distance requirement (3) can be expressed as the minimization of the following normalized distance: $$\begin{equation*} d_{n} = d/a = \Vert \mathbf {s}- \mathbf {t}\Vert + \Vert \mathbf {s}- \mathbf {r}\Vert. \tag{12} \end{equation*}$$ In summary, the specular reflection point $\mathbf {s}$ is governed by (10), (11), and the minimal distance in (12).

B. Characterization of the Specular Reflection Point

In this subsection, a characterization of the specular reflection point will be discussed. A necessary condition for the distance to be minimal is that the partial derivative of the distance with respect to $\mathbf {s}$ is zero. As the vector $\mathbf {s}$ is required to satisfy the constraint (10), the following Hamiltonian is considered: $$\begin{equation*} J = \Vert \mathbf {s}- \mathbf {t}\Vert + \Vert \mathbf {s}- \mathbf {r}\Vert + \lambda \, (\mathbf {s}^{T} \, \mathbf {S}^{-1} \, \mathbf {s}- 1) \tag{13} \end{equation*}$$ where $\lambda$ is the Lagrange multiplier. The necessary conditions for optimality are given by $\frac{\partial J}{\partial \lambda } = 0$ and $\frac{\partial J}{\partial \mathbf {s}} = 0$. The former leads to (10), and the latter results in $$\begin{equation*} \frac{\partial J}{\partial \mathbf {s}} = \frac{\mathbf {s}- \mathbf {t}}{\Vert \mathbf {s}- \mathbf {t}\Vert } + \frac{\mathbf {s}- \mathbf {r}}{\Vert \mathbf {s}- \mathbf {r}\Vert } + 2 \lambda \, \mathbf {S}^{-1} \, \mathbf {s}= 0. \tag{14} \end{equation*}$$ Multiplying (14) from left by $\mathbf {s}^{T}$ and applying (10), the Lagrange multiplier can then be solved as $\lambda = - \frac{1}{2} \mathbf {s}^{T} (\frac{\mathbf {s}- \mathbf {t}}{\Vert \mathbf {s}- \mathbf {t}\Vert } + \frac{\mathbf {s}- \mathbf {r}}{\Vert \mathbf {s}- \mathbf {r}\Vert })$. Substituting this expression into (14) yields a criterion on the specular reflection point $$\begin{equation*} \left(\mathbf {I}- \mathbf {S}^{-1} \, \mathbf {s}\mathbf {s}^{T} \right) \left(\frac{\mathbf {s}- \mathbf {t}}{\Vert \mathbf {s}- \mathbf {t}\Vert } + \frac{\mathbf {s}- \mathbf {r}}{\Vert \mathbf {s}- \mathbf {r}\Vert } \right) = 0 \tag{15} \end{equation*}$$ where $\mathbf {I}$ is the identity matrix. Combining (11) and (15), one can remove the dependence on the terms $\Vert \mathbf {s}- \mathbf {t}\Vert$ and $\Vert \mathbf {s}- \mathbf {r}\Vert$ to yield $$\begin{equation*} \left(\mathbf {I}- \mathbf {S}^{-1} \, \mathbf {s}\mathbf {s}^{T} \right) \left(\frac{\mathbf {s}- \mathbf {t}}{1 - \mathbf {s}^{T} \, \mathbf {S}^{-1} \, \mathbf {t}} + \frac{\mathbf {s}- \mathbf {r}}{1 - \mathbf {s}^{T} \, \mathbf {S}^{-1} \, \mathbf {r}} \right) = 0. \tag{16} \end{equation*}$$ Some manipulations reveal that (16) can be written as $$\begin{equation*} \mathbf {f}(\mathbf {s}) = 0 \tag{17} \end{equation*}$$ where $\mathbf {f}(\mathbf {s})$ is defined in (18) [Eq. (18) shown at the bottom of this page]. $$\begin{align*} \mathbf {f}(\mathbf {s}) = &(1 - \mathbf {s}^{T} \mathbf {S}^{-1} \mathbf {r}) (\mathbf {s}- \mathbf {t}) + (1 - \mathbf {s}^{T} \mathbf {S}^{-1} \mathbf {t}) (\mathbf {s}- \mathbf {r})- \mathbf {s}^{T} \left((1 - \mathbf {s}^{T} \mathbf {S}^{-1} \mathbf {r}) (\mathbf {s}- \mathbf {t}) + (1 - \mathbf {s}^{T} \mathbf {S}^{-1} \mathbf {t}) (\mathbf {s}- \mathbf {r}) \right) \mathbf {S}^{-1} \mathbf {s} \tag{18} \end{align*}$$ Let $\alpha = 1 - \mathbf {s}^{T} \mathbf {S}^{-1} \mathbf {r}$, $\beta = 1 - \mathbf {s}^{T} \mathbf {S}^{-1} \mathbf {t}$, and $\gamma = \mathbf {s}^{T} ((1 - \mathbf {s}^{T} \mathbf {S}^{-1} \mathbf {r}) (\mathbf {s}- \mathbf {t}) + (1 - \mathbf {s}^{T} \mathbf {S}^{-1} \mathbf {t}) (\mathbf {s}- \mathbf {r}))$; (17) can indeed be written as $$\begin{equation*} \alpha \, (\mathbf {s}- \mathbf {t}) + \beta \, (\mathbf {s}- \mathbf {r}) = \gamma \, \mathbf {S}^{-1} \, \mathbf {s}. \tag{19} \end{equation*}$$ Hence, the three vectors $\mathbf {s}- \mathbf {t}$, $\mathbf {s}- \mathbf {r}$, and $\mathbf {S}^{-1} \mathbf {s}$ are linearly dependent or in the same plane. Equation (19) depicts the geometric relationship between the specular reflection point $\mathbf {s}$ with respect to the transmitter at $\mathbf {t}$, the receiver at $\mathbf {r}$, and the ellipsoid governed by $\mathbf {S}$. The coefficients in (19) depend on the unknown $\mathbf {s}$ in which $\alpha$ and $\beta$ are linear function of $\mathbf {s}$, while $\gamma$ is a cubic function of $\mathbf {s}$. Thus, algebraically, the determination of the specular reflection point can be stated as the solving of (17), which is a set of three quartic polynomial equations that involve three unknown variables.

Without loss of generality, the variable $\gamma$ is assumed to be nonzero. Indeed, if $\gamma$ is zero, (19) implies that $\mathbf {t}$, $\mathbf {r}$, and $\mathbf {s}$ are along the same line, and the determination of the specular reflection point is trivial.

Equation (19) can be used to determine the specular reflection in an iterative manner. More precisely, let $\mathbf {s}_{k}$ be the estimate of the specular reflection point at the $k$th iteration; one can then compute the coefficients at the iteration as $\alpha _{k} = 1 - \mathbf {s}_{k}^{T} \mathbf {S}^{-1} \mathbf {r}$, $\beta _{k} = 1 - \mathbf {s}_{k}^{T} \mathbf {S}^{-1} \mathbf {t}$, and $\gamma _{k} = \mathbf {s}_{k}^{T} (\alpha _{k} \, (\mathbf {s}_{k} - \mathbf {t}) + \beta _{k} (\mathbf {s}_{k} - \mathbf {r}))$. The estimate at the next iteration can be formed from (19) as $\mathbf {s}_{k+1} = \gamma _{k}^{-1} \mathbf {S}(\alpha _{k} \, (\mathbf {s}_{k} - \mathbf {t}) + \beta _{k} (\mathbf {s}_{k} - \mathbf {r}))$. Note that this iteration scheme does not involve the computation of the norm. The convergence of the iterative approach, however, depends on the selection of the initial estimate.

C. Decoupling Approach

Note that (17) that characterizes the specular reflection point $\mathbf {s}$ is a vector equation of dimension 3. In the following, an effort is made to decouple the vector equation to provide insights of the problem. The reflection path is from $\mathbf {t}$ to $\mathbf {s}$ and then to $\mathbf {r}$. Thus, the three vectors $\mathbf {t}$, $\mathbf {r}$, and $\mathbf {s}$ span a plane, and the reflection is manifested on the ellipse that is the intersection of the plane and the ellipsoid. A normal vector of the plane can be obtained as $\mathbf {m}= (\mathbf {t}- \mathbf {s}) \times (\mathbf {r}- \mathbf {s})$. In the $\lbrace \mathbf {n}_{1}, \mathbf {n}_{2}, \mathbf {n}_{3} \rbrace$ coordinates, the normal vector is $$\begin{equation*} \mathbf {m}= \left[ \begin{array}{c}r_{2} \, s_{3} \\ (t_{1} - r_{1}) \, s_{3} \\ t_{1} \, r_{2} - r_{2} \, s_{1} - (t_{1} - r_{1}) \, s_{2} \end{array} \right] = \mathbf {K}\, \mathbf {s}+ \mathbf {h} \tag{20} \end{equation*}$$ where $\mathbf {K}= \left[ \scriptsize\begin{array}{ccc}0 & 0 & r_{2} \\ 0 & 0 & t_{1} - r_{1} \\ -r_{2} & -(t_{1} - r_{1}) & 0 \end{array} \right]$ and $\mathbf {h}= \mathbf {t}\times \mathbf {r}= \left[ \begin{array}{ccc}0 & 0 & t_{1} r_{2} \end{array} \right]^{T}$. It is noted that $\mathbf {m}^{T} \mathbf {t}= \mathbf {m}^{T} \mathbf {r}= \mathbf {m}^{T} \mathbf {s}= t_{1} r_{2} s_{3}$, implying that $\mathbf {m}$ is normal to the plane spanned by $\mathbf {t}$, $\mathbf {r}$, and $\mathbf {s}$. The matrix $\mathbf {K}$ is a skew-symmetric matrix. Thus, one can express $\mathbf {K}\mathbf {s}$ as $\mathbf {K}\mathbf {s}= \mathbf {k}\times \mathbf {s}$, where $\mathbf {k}= \left[ \begin{array}{ccc}-t_{1}+r_{1} & r_{2} & 0 \end{array} \right]^{T} = \mathbf {r}- \mathbf {t}$. We may, hereafter, use the notation $[\mathbf {k}\times ]$ to denote the skew-symmetric matrix formed from the vector $\mathbf {k}$. It is further observed that the vector $\mathbf {k}$ is orthogonal to $\mathbf {h}$.

Let $\mathbf {Q}= [ \begin{array}{ccc}\gamma ^{-1}\mathbf {s}& -\gamma ^{-1}\mathbf {m}& \mathbf {S}(\mathbf {s}\times \mathbf {h}) \end{array} ]$. If $\mathbf {Q}$ is nonsingular, then $\mathbf {f}(\mathbf {s}) = 0$ is equivalent to $\mathbf {Q}^{T} \mathbf {f}(\mathbf {s}) = 0$. The first element of $\mathbf {Q}^{T} \mathbf {f}(\mathbf {s})$ is obtained by multiplying $\mathbf {f}(\mathbf {s})$ from left by $\gamma ^{-1}\mathbf {s}^{T}$, resulting in $$\begin{equation*} \gamma ^{-1} \mathbf {s}^{T} \mathbf {f}(\mathbf {s}) = 1- \mathbf {s}^{T} \mathbf {S}^{-1} \mathbf {s}= 0. \tag{21} \end{equation*}$$ This essentially recovers (10). It is observed that the second column of $\mathbf {Q}$ is orthogonal to $\mathbf {s}- \mathbf {t}$ and $\mathbf {s}- \mathbf {r}$. Thus, the second element of $\mathbf {Q}^{T} \mathbf {f}(\mathbf {s})$ is not affected by the first two terms of $\mathbf {f}(\mathbf {s})$ in (18), and the following expression is obtained: $$\begin{equation*} -\gamma ^{-1} \mathbf {m}^{T} \mathbf {f}(\mathbf {s}) = \mathbf {m}^{T} \mathbf {S}^{-1} \mathbf {s}= \mathbf {s}^{T} \mathbf {K}^{T} \mathbf {S}^{-1} \mathbf {s}+ \mathbf {h}^{T} \mathbf {S}^{-1} \mathbf {s}= 0. \tag{22} \end{equation*}$$ Furthermore, the third column of $\mathbf {Q}$ is orthogonal to $\mathbf {S}^{-1} \mathbf {s}$. Hence, the third element of $\mathbf {Q}^{T} \mathbf {f}(\mathbf {s})$ is $$\begin{align*} (\mathbf {s}\times \mathbf {h})^{T} \mathbf {S}\mathbf {f}(\mathbf {s}) = & (1- \mathbf {s}^{T} \mathbf {S}^{-1} \mathbf {r}) (\mathbf {s}\times \mathbf {h})^{T} \mathbf {S}(\mathbf {s}- \mathbf {t}) \\ & + (1- \mathbf {s}^{T} \mathbf {S}^{-1} \mathbf {t}) (\mathbf {s}\times \mathbf {h})^{T} \mathbf {S}(\mathbf {s}- \mathbf {r}) \\ = & 0. \tag{23} \end{align*}$$ Therefore, if the matrix $\mathbf {Q}$ is nonsingular, the determination of the specular reflection point becomes the problem of solving three simultaneous polynomial equations (21)–​(23) of three variables. Two polynomials (21) and (22) are quadratic, and the third polynomial (23) is cubic. This is considered much simpler than the original formulation in (17).

D. Spherical Earth

The above characterization of the specular reflection point in terms of (21)–​(23) is further investigated to establish a computational scheme. To this end, the spherical Earth is first considered in which the matrix $\mathbf {E}$ in (1) is assumed to be a diagonal matrix $\mathbf {E}= a^{2} \, \mathbf {I}$, where $a$ is the radius. As a result, the matrix $\mathbf {S}$ becomes the identity matrix, and (21) is reduced to $$\begin{equation*} \mathbf {s}^{T} \mathbf {s}= \mathbf {I}. \tag{24} \end{equation*}$$ Observe that the term $\mathbf {s}^{T} \mathbf {K}^{T} \mathbf {S}^{-1} \mathbf {s}$ in (22) is zero as the matrix $\mathbf {K}$ is skew symmetric. Thus, (22) becomes $\mathbf {h}^{T} \mathbf {s}= 0$, which implies that the third element of $\mathbf {s}$ is zero $$\begin{equation*} s_{3} = 0. \tag{25} \end{equation*}$$ As a result, the determination of the specular reflection point becomes a problem with two unknown variables $s_{1}$ and $s_{2}$. It can further be shown that the term $(\mathbf {s}\times \mathbf {h})^{T} \mathbf {S}\mathbf {s}$ in (23) is zero; thus, (23) can be simplified as $$\begin{equation*} (1- \mathbf {s}^{T} \mathbf {r}) (\mathbf {s}\times \mathbf {h})^{T} \mathbf {t}+ (1- \mathbf {s}^{T} \mathbf {t}) (\mathbf {s}\times \mathbf {h})^{T} \mathbf {r}= 0. \tag{26} \end{equation*}$$ In terms of components $s_{1}$ and $s_{2}$, (24) and (26) are given, respectively, by $$\begin{equation*} s_{1}^{2} + s_{2}^{2} - 1 = 0 \tag{27} \end{equation*}$$ and $$\begin{equation*} \left(1 - (r_{1} s_{1} + r_{2} s_{2}) \right) t_{1} s_{2} + \left(1 - t_{1} s_{1} \right) (r_{1} s_{2} - r_{2} s_{1}) = 0. \tag{28} \end{equation*}$$ To solve the two quadratic polynomial equations with two variables, one can employ the methods of Bezout determinant and resultant of two polynomials to eliminate one variable [21]. More precisely, the condition for the two quadratic polynomials $a_{2}(s_{1}) \, s_{2}^{2} + a_{1}(s_{1}) \, s_{2} + a_{0}(s_{1}) = 0$ and $b_{2}(s_{1}) \, s_{2}^{2} + b_{1}(s_{1}) \, s_{2} + b_{0}(s_{1}) = 0$ to have a common root is [22] $$\begin{equation*} \left(a_{2} b_{1} - a_{1} b_{2} \right) \left(a_{1} b_{0} - a_{0} b_{1} \right) - \left(a_{2} b_{0} - a_{0} b_{2} \right)^{2} = 0. \tag{29} \end{equation*}$$ Applying the above result to (27) and (28) yields a quartic polynomial equation of $s_{1}$ in (30). Therefore, by solving (30), one can determine $s_{1}$ and, consequently, compute $s_{2}$ by using (27) and (28) through the back substitution process. It is noted that there may be four candidate solutions to (30), which means that the simultaneous equations (27) and (28) may have four solutions. The correct solution can be resolved by resorting to the minimal distance property. Thus, the determination of the specular reflection point for a spherical Earth amounts to solving a fourth-order polynomial equation.

It is further remarked that the matrix $\mathbf {Q}$ in the spherical Earth case is nonsingular. Indeed, the matrix $\mathbf {Q}$ in this case can be expressed as $$\begin{equation*} \mathbf {Q}= \left[ \begin{array}{ccc}\mathbf {s}& \mathbf {m}& \mathbf {s}\times \mathbf {h}\end{array} \right] \, \text{diag} \lbrace \gamma ^{-1}, - \gamma ^{-1}, 1 \rbrace. \end{equation*}$$ Thus, $\mathbf {Q}$ is nonsingular if and only if $[ \begin{array}{ccc}\mathbf {s}& \mathbf {m}& \mathbf {s}\times \mathbf {h}\end{array} ]$ is nonsingular. Note that the two vectors $\mathbf {s}$ and $\mathbf {s}\times \mathbf {h}$ are in the $\mathbf {n}_{1}$–$\mathbf {n}_{2}$ plane and orthogonal to each other. The vector $\mathbf {m}$ is along the direction of $\mathbf {n}_{3}$. Therefore, $\mathbf {Q}$ is nonsingular. [Eq. (30) shown at the bottom of this page]. $$\begin{align*} (s_{1}^{2} - 1) (t_{1} + r_{1} - 2 t_{1} r_{1} s_{1}) ^{2} + \left(r_{2} s_{1} (-1 + t_{1} s_{1})+ (s_{1}^{2} - 1) t_{1} r_{2} \right)^{2} = 0 \tag{30} \end{align*}$$

E. Ellipsoidal Earth

The resultant method is investigated in this subsection to provide a computational procedure of the specular reflection point with respect to the ellipsoidal Earth. Let ${\boldsymbol{\Sigma }}= \text{diag} \lbrace 1, 1, b/a \rbrace$; then, $\mathbf {S}$ can be expressed as $\mathbf {S}= \mathbf {N}^{T} {\boldsymbol{\Sigma }}^{2} \mathbf {N}$ and $\mathbf {S}^{-1}$ becomes $\mathbf {S}^{-1} = \mathbf {N}^{T} {\boldsymbol{\Sigma }}^{-2} \mathbf {N}$. Let $\mathbf {z}= {\boldsymbol{\Sigma }}^{-1} \mathbf {N}\mathbf {s}= [ \begin{array}{ccc}z_{1} & z_{2} & z_{3} \end{array}]^{T}$; then, (21) can be written as $$\begin{equation*} \mathbf {z}^{T} \, \mathbf {z}= 1 \tag{31} \end{equation*}$$ or $$\begin{equation*} z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 1. \tag{32} \end{equation*}$$ In addition, (22) can also be expressed in terms of $\mathbf {z}$ as $$\begin{equation*} - \mathbf {z}^{T} \, {\boldsymbol{\Sigma }}\, [(\mathbf {N}\mathbf {k}) \times ] \, {\boldsymbol{\Sigma }}^{-1} \, \mathbf {z}+ \mathbf {z}_{h}^{T} \, \mathbf {z}= 0 \tag{33} \end{equation*}$$ where $\mathbf {z}_{h} = {\boldsymbol{\Sigma }}^{-1} \mathbf {N}\mathbf {h}$. Furthermore, let $\mathbf {z}_{t} = {\boldsymbol{\Sigma }}^{-1} \mathbf {N}\mathbf {t}$ and $\mathbf {z}_{r} = {\boldsymbol{\Sigma }}^{-1} \mathbf {N}\mathbf {r}$; then, (23) can be restated as $$\begin{align*} & (1-\mathbf {z}^{T} \mathbf {z}_{r}) \, \mathbf {z}^{T} {\boldsymbol{\Sigma }}^{-1} \, [({\boldsymbol{\Sigma }}^{2} \mathbf {N}\mathbf {h}) \times ] \, {\boldsymbol{\Sigma }}(\mathbf {z}- \mathbf {z}_{t}) \\ &\quad + (1-\mathbf {z}^{T} \mathbf {z}_{t}) \, \mathbf {z}^{T} {\boldsymbol{\Sigma }}^{-1} \, [({\boldsymbol{\Sigma }}^{2} \mathbf {N}\mathbf {h}) \times ] \, {\boldsymbol{\Sigma }}(\mathbf {z}- \mathbf {z}_{r}) \\ = & 0. \tag{34} \end{align*}$$

It is noted that both (33) and (34) contain a term that bears a form as a product of a diagonal matrix, skew-symmetric matrix, and the inverse of the diagonal matrix, making it possible to have a simple expression. More precisely, let $\mathbf {N}\mathbf {k}= \left[ \begin{array}{ccc}k_{1} & k_{2} & k_{3} \end{array} \right]$; then, the term $\mathbf {z}^{T} \, {\boldsymbol{\Sigma }}\, [(\mathbf {N}\mathbf {k}) \times ] \, {\boldsymbol{\Sigma }}^{-1} \, \mathbf {z}$ can be expressed as $$\begin{equation*} \mathbf {z}^{T} \, {\boldsymbol{\Sigma }}\, [(\mathbf {N}\mathbf {k}) \times ] \, {\boldsymbol{\Sigma }}^{-1} \, \mathbf {z}= k_{1} \, \delta \, z_{2} z_{3} - k_{2} \, \delta \, z_{1} z_{3} \end{equation*}$$ where $\delta = b/a - a/b$. That is, the quadratic function is indeed a bilinear function. As a result, (33) can be rewritten in component form as $$\begin{equation*} h_{1} z_{1} + h_{2} z_{2} + h_{3} z_{3} - k_{1} \, \delta \, z_{2} z_{3} + k_{2} \, \delta \, z_{1} z_{3} = 0 \tag{35} \end{equation*}$$ where $h_{1}$, $h_{2}$, and $h_{3}$ are components of the vector $\mathbf {z}_{h}$. In a similar manner, let $\mathbf {z}_{t} = [ \begin{array}{ccc}\overline{t}_{1} & \overline{t}_{2} & \overline{t}_{3} \end{array} ]^{T}$ and $\mathbf {z}_{r} = [ \begin{array}{ccc}\overline{r}_{1} & \overline{r}_{2} & \overline{r}_{3} \end{array} ]^{T}$; then, (34) becomes (36) in which $f_{1} = (b/a)^{3} h_{3} \overline{t}_{2} - (b/a) h_{2} \overline{t}_{3}$, $f_{2} = - (b/a)^{3} h_{3} \overline{t}_{1} + (b/a) h_{1} \overline{t}_{3}, f_{3} = (a/b) (h_{2} \overline{t}_{1} - h_{1} \overline{t}_{2}), g_{1} = (b/a)^{3} h_{3} \overline{r}_{2} - (b/a) h_{2} \overline{r}_{3}$, $g_{2} = - (b/a)^{3} h_{3} \overline{r}_{1} + (b/a) h_{1} \overline{r}_{3}$, and $g_{3} = (a/b) (h_{2} \overline{r}_{1} - h_{1} \overline{r}_{2})$. [Eq. (36) shown at the bottom of this page] $$\begin{align*} (\! 1 {-}\overline{r}_{1} z_{1} {-} \overline{r}_{2} z_{2} {-} \overline{r}_{3} z_{3}\!) (f_{1} z_{1} {+} f_{2} z_{2} {+} f_{3} z_{3} {+} h_{2} \delta z_{1} z_{3} {-} h_{1} \delta z_{2} z_{3}\!) {+} (\! 1{-}\overline{t}_{1} z_{1} {-} \overline{t}_{2} z_{2} {-} \overline{t}_{3} z_{3}\!) (g_{1} z_{1} {+} g_{2} z_{2} {+} g_{3} z_{3} {+} h_{2} \delta z_{1} z_{3} {-} h_{1} \delta z_{2} z_{3}\!) {=} 0 \tag{36} \end{align*}$$

With respect to the three polynomial equations (32), (35), and (36), one can then employ resultants to eliminate variables one by one. Note that (36) can be expressed as $\overline{b}_{1}(z_{1}, z_{2}) \, z_{3} + \overline{b}_{0}(z_{1}, z_{2}) = 0$ for some $\overline{b}_{1}(z_{1}, z_{2})$ and $\overline{b}_{0}(z_{1}, z_{2})$. Also, both (32) and (36) can be cast as $\overline{a}_{2}(z_{1}, z_{2}) \, z_{3}^{2} + \overline{a}_{1}(z_{1}, z_{2}) \, z_{3} + \overline{a}_{0}(z_{1}, z_{2}) = 0$ for some $\overline{a}_{2}(z_{1}, z_{2})$, $\overline{a}_{1}(z_{1}, z_{2})$, and $\overline{a}_{0}(z_{1}, z_{2})$. Then, the condition for the existence of common solution $z_{3}$ is [21], [22] $$\begin{equation*} \overline{a}_{2} \, \overline{b}_{0}^{2} + \overline{a}_{0} \, \overline{b}_{1}^{2}- \overline{a}_{1} \, \overline{b}_{0} \overline{b}_{1} = 0. \tag{37} \end{equation*}$$ Applying this result to (32) and (35) yields (38). Similarly, applying the result to (32) and (35) leads to (39) in which $\Delta (z_{1}, z_{2})$ is defined as in (40). As a result, the variable $z_{3}$ is eliminated, and the resulting (38) and (39) are quartic polynomials of $z_{1}$ and $z_{2}$. The resultant of the two polynomials can then be evaluated. By eliminating $z_{2}$, the resultant is a polynomial of $z_{1}$ with a generic degree of 16. Once the polynomial equation of $z_{1}$ is solved, the other variables $z_{2}$ and $z_{3}$ can be determined by back substitution. Consequently, the specular reflection point is obtained as $\mathbf {s}= \mathbf {N}^{T} {\boldsymbol{\Sigma }}\, \mathbf {z}$. Thereafter, the specular reflection point in the ECEF frame can be computed as in (9) or, more precisely, $\mathbf {p}_{s} = a \, {\boldsymbol{\Sigma }}\, \mathbf {z}$. In summary, the specular reflection point can be characterized in terms of a set of three polynomials with three variables. By using the resultant technique, the key computation step is the solving of a polynomial equation with one variable of generic degree 16. The procedure for the conversion of three polynomials to one polynomial is routine yet tedious. Tools of symbolic processing, such as [23], can be used to facilitate the tasks. [Equ. (38--40) shown at the bottom of the next page] $$\begin{align*} 0 = &(h_{1} z_{1} + h_{2} z_{2})^{2} + (z_{1}^{2} + z_{2}^{2} - 1) (h_{3} - k_{1} \delta z_{2} + k_{2} \delta z_{1}) ^{2} \tag{38} \\ 0 = & \left(r_{3} (f_{3} \!+\! h_{2} \delta z_{1} \!-\! h_{1} \delta z_{2}) \!+\! t_{3} (g_{3} + h_{2} \delta z_{1} \!-\! h_{1} \delta z_{2}) \right)(h_{1} z_{1} \!+\! h_{2} z_{2})^{2} \!-\! \Delta (z_{1}, z_{2}) (h_{3} \!+\! k_{2} \delta z_{1} \!-\! k_{1} \delta z_{2}) (h_{1} z_{1} \!+\! h_{2} z_{2}) \\ &- \left((1 - \overline{r}_{1} z_{1} - \overline{r}_{2} z_{2}) (f_{1} z_{1} + f_{2} z_{2})+ (1 - \overline{t}_{1} z_{1} - \overline{t}_{2} z_{2}) (g_{1} z_{1} + g_{2} z_{2}) \right) (h_{3} + k_{2} \delta z_{1} - k_{1} \delta z_{2})^{2} \tag{39}\\ \Delta (z_{1},z_{2}) =& \overline{r}_{3} (f_{1} z_{1} {+} f_{2} z_{2}) {-} (1 {-} \overline{r}_{1} z_{1} {-} \overline{r}_{2} z_{2})(f_{3} {+} h_{2} \delta z_{1} {-} h_{1} \delta z_{2}) {+} \overline{t}_{3} (g_{1} z_{1} {+} g_{2} z_{2}) {-} (1 {-} \overline{t}_{1} z_{1} {-} \overline{t}_{2} z_{2})(g_{3} {+} h_{2} \delta z_{1} {-} h_{1} \delta z_{2}) \tag{40} \end{align*}$$

SECTION III. Sensitivity Analysis

Let $\mathbf {p}_{t}$ and $\mathbf {p}_{r}$ be the nominal positions of the transmitter and the receiver, respectively; then, the aforementioned algorithm can be applied to determine the nominal position of the specular reflection point $\mathbf {p}_{s}$. In practice, the positions of the transmitter and the receiver are subject to perturbations, leading to deviations on the estimate of the specular reflection point. In this section, sensitivity analysis is performed to quantify the effect of the uncertainties.

It is assumed that the positions $\mathbf {p}_{t}$ and $\mathbf {p}_{r}$ are subject to additive perturbation to become $\mathbf {p}_{t} + \Delta _{t}$ and $\mathbf {p}_{s} + \Delta _{r}$, respectively. Thus, the resulting specular reflection point deviates from $\mathbf {p}_{s}$ to $\mathbf {p}_{s} + \Delta _{s}$ for some $\Delta _{s}$. It is desired to assess the sensitivity of $\Delta _{s}$ with respect to $\Delta _{t}$ and $\Delta _{r}$. In the following, a perturbation approach is developed to assess the sensitivity.

Recall that the specular reflection point $\mathbf {p}_{s}$ is characterized as in (3). Let $\lambda$ be a Lagrange multiplier; the Hamiltonian of the optimization problem is $$\begin{equation*} J = \Vert \mathbf {p}_{t} - \mathbf {p}_{s} \Vert + \Vert \mathbf {p}_{s} - \mathbf {p}_{r} \Vert + \lambda \left(\mathbf {p}_{s}^{T} \mathbf {E}^{-1} \mathbf {p}_{s} - 1 \right). \tag{41} \end{equation*}$$ This objective function is similar to (13) except that it is formulated in the original ECEF coordinates. Following a similar approach, the Lagrange multiplier is solved as $\lambda = - \frac{1}{2} \mathbf {p}_{s}^{T} (\frac{\mathbf {p}_{s} - \mathbf {p}_{t}}{\Vert \mathbf {p}_{s} - \mathbf {p}_{t} \Vert } + \frac{\mathbf {p}_{s} - \mathbf {p}_{r}}{\Vert \mathbf {p}_{s} - \mathbf {p}_{r} \Vert })$ and the specular reflection point $\mathbf {p}_{s}$ is characterized by $$\begin{equation*} \left(\mathbf {I}- \mathbf {E}^{-1} \mathbf {p}_{s} \mathbf {p}_{s}^{T} \right) \left(\frac{\mathbf {p}_{s} - \mathbf {p}_{t}}{\Vert \mathbf {p}_{s} - \mathbf {p}_{t} \Vert } + \frac{\mathbf {p}_{s} - \mathbf {p}_{r}}{\Vert \mathbf {p}_{s} - \mathbf {p}_{r} \Vert } \right) = 0. \tag{42} \end{equation*}$$

Notice that if a vector $\mathbf {q}$ is perturbed as $\mathbf {q}+ \Delta _{q}$ and the perturbation is relatively small, then the quantity $\frac{1}{\Vert \mathbf {q}\Vert }$ is perturbed as $\frac{1}{\Vert \mathbf {q}+ \Delta _{q} \Vert } \approx \frac{1}{\Vert \mathbf {q}\Vert } -\frac{1}{\Vert \mathbf {q}\Vert ^{3}} \mathbf {q}^{T} \Delta _{q}$. Furthermore, the term $\frac{\mathbf {q}+ \Delta _{q}}{\Vert \mathbf {q}+ \Delta _{q} \Vert }$ can be approximated as $\frac{\mathbf {q}}{\Vert \mathbf {q}\Vert } + \frac{1}{\Vert \mathbf {q}\Vert } (\mathbf {I}- (\frac{\mathbf {q}}{\Vert \mathbf {q}\Vert }) (\frac{\mathbf {q}}{\Vert \mathbf {q}\Vert })^{T}) \Delta _{q}$. Applying the perturbation analysis to (42) and by keeping the first-order terms, it can be shown that $\Delta _{s}$ is related to $\Delta _{t}$ and $\Delta _{r}$ as follows: $$\begin{equation*} \mathbf {A}_{s} \, \Delta _{s} = \mathbf {A}_{t} \, \Delta _{t} + \mathbf {A}_{r} \Delta _{r} \tag{43} \end{equation*}$$ where $\mathbf {A}_{t}$, $\mathbf {A}_{r}$, and $\mathbf {A}_{s}$ are defined in (44). [Eq. (44) shown at the bottom of the next page] $$\begin{align*} \mathbf {A}_{t} = & \frac{1}{\Vert \mathbf {p}_{s} - \mathbf {p}_{t} \Vert } (\mathbf {I}- \mathbf {E}^{-1} \mathbf {p}_{s} \mathbf {p}_{s}^{T})\left(\mathbf {I}- \left(\frac{\mathbf {p}_{s} - \mathbf {p}_{t}}{\Vert \mathbf {p}_{s} - \mathbf {p}_{t} \Vert }\right) \left(\frac{\mathbf {p}_{s} - \mathbf {p}_{t}}{\Vert \mathbf {p}_{s} - \mathbf {p}_{t} \Vert }\right)^{T} \right) \\ \mathbf {A}_{r} = & \frac{1}{\Vert \mathbf {p}_{s} - \mathbf {p}_{r} \Vert } (\mathbf {I}- \mathbf {E}^{-1} \mathbf {p}_{s} \mathbf {p}_{s}^{T})\left(\mathbf {I}- \left(\frac{\mathbf {p}_{s} - \mathbf {p}_{r}}{\Vert \mathbf {p}_{s} - \mathbf {p}_{r} \Vert }\right) \left(\frac{\mathbf {p}_{s} - \mathbf {p}_{r}}{\Vert \mathbf {p}_{s} - \mathbf {p}_{r} \Vert }\right)^{T} \right) \\ \mathbf {A}_{s} = & \mathbf {A}_{t} + \mathbf {A}_{r} - \mathbf {E}^{-1} \left(\mathbf {p}_{s}^{T} \left(\frac{\mathbf {p}_{s} - \mathbf {p}_{t}}{\Vert \mathbf {p}_{s} - \mathbf {p}_{t} \Vert } + \frac{\mathbf {p}_{s} - \mathbf {p}_{r}}{\Vert \mathbf {p}_{s} - \mathbf {p}_{r} \Vert }\right) \mathbf {I}+ \mathbf {p}_{s} \left(\frac{\mathbf {p}_{s} - \mathbf {p}_{t}}{\Vert \mathbf {p}_{s} - \mathbf {p}_{t} \Vert } + \frac{\mathbf {p}_{s} - \mathbf {p}_{r}}{\Vert \mathbf {p}_{s} - \mathbf {p}_{r} \Vert }\right) ^{T} \right) \tag{44} \end{align*}$$

Assume that $\Delta _{t}$ and $\Delta _{r}$ are independent zero-mean random vectors with covariance matrices $\sigma _{t}^{2} \, \mathbf {I}$ and $\sigma _{r}^{2} \, \mathbf {I}$, respectively. From (43), the covariance matrix of $\Delta _{s}$ is $$\begin{equation*} \mathbb {E}\lbrace \Delta _{s} \Delta _{s}^{T} \rbrace = \mathbf {H}_{s}^{T} \, \left[ \begin{array}{cc}\mathbf {A}_{t} & \mathbf {A}_{r} \end{array} \right] \, \left[ \begin{array}{cc}\sigma _{t}^{2} \mathbf {I}& 0 \\ 0 & \sigma _{r}^{2} \mathbf {I}\end{array} \right] \, \left[ \begin{array}{c}\mathbf {A}_{t}^{T} \\ \mathbf {A}_{r}^{T} \end{array} \right] \, \mathbf {H}_{s} \tag{45} \end{equation*}$$ where $$\begin{equation*} \mathbf {H}_{s} = \mathbf {A}_{s} (\mathbf {A}_{s}^{T} \mathbf {A}_{s})^{-1}. \end{equation*}$$ Furthermore, the mean square error is given by $$\begin{align*} \mathbb {E}\lbrace \Delta _{s}^{T} \Delta _{s} \rbrace = &\text{trace} (\mathbf {H}_{s}^{T} \mathbf {A}_{t} \mathbf {A}_{t}^{T} \mathbf {H}_{s}) \, \sigma _{t}^{2} \\ &+ \text{trace}(\mathbf {H}_{s}^{T} \mathbf {A}_{r} \mathbf {A}_{r}^{T} \mathbf {H}_{s}) \, \sigma _{r}^{2}. \tag{46} \end{align*}$$ Define the dilution of precision in reflection due to transmitter (DOPR$_{t}$) as $$\begin{equation*} \text{DOPR}_{t} = \sqrt{ \text{trace}(\mathbf {H}_{s}^{T} \mathbf {A}_{t} \mathbf {A}_{t}^{T} \mathbf {H}_{s}) } \tag{47} \end{equation*}$$ and the dilution of precision in reflection due to receiver (DOPR$_{r}$) as $$\begin{equation*} \text{DOPR}_{r} = \sqrt{ \text{trace} (\mathbf {H}_{s}^{T} \mathbf {A}_{r} \mathbf {A}_{r}^{T} \mathbf {H}_{s}) }. \tag{48} \end{equation*}$$ These dilutions of precision play the role in quantifying the deviation of the transmitter/receiver on the resulting position error of the reflection point. From (46), the variance of the reflection point position error $\sigma _{s}^{2} = \mathbb {E}\lbrace \Delta _{s}^{T} \Delta _{s} \rbrace$ can be expressed as $$\begin{equation*} \sigma _{s}^{2} = (\text{DOPR}_{t})^{2} \cdot \sigma _{t}^{2} + (\text{DOPR}_{r})^{2} \cdot \sigma _{r}^{2} . \tag{49} \end{equation*}$$

In the above equation, it is assumed that the errors on $\mathbf {p}_{t}$ and $\mathbf {p}_{r}$ are uncorrelated. In practice, some receivers being at a lower orbit utilize GNSS signals for positioning. Under this circumstance, the receiver position error is correlated with the transmitter position error, and the following model can be used: $$\begin{equation*} \sigma _{r}^{2} = \text{PDOP}^{2} \sigma _{t}^{2} + \sigma _{m}^{2} \end{equation*}$$ where PDOP is the position dilution of precision [17] and $\sigma _{m}^{2}$ is the variance associated with the errors in GNSS signal propagation and GNSS receiver noise. Consequently $$\begin{align*} \sigma _{s}^{2} =& \left((\text{DOPR}_{t})^{2} + (\text{DOPR}_{r})^{2} \cdot (\text{PDOP})^{2} \right)\\ & \times\sigma _{t}^{2} + (\text{DOPR}_{r})^{2} \cdot \sigma _{m}^{2}. \end{align*}$$

As the specular reflection point is constrained on the ellipsoidal Earth, it amounts to analyzing the deviation in the local horizontal plane. To this end, left multiplying (43) by $\mathbf {p}_{s}^{T}$ results in $\mathbf {p}_{s}^{T}\mathbf {A}_{s} \Delta _{s} = 0$ or $(\frac{\mathbf {p}_{s} - \mathbf {p}_{t}}{\Vert \mathbf {p}_{s} - \mathbf {p}_{t} \Vert } + \frac{\mathbf {p}_{s} - \mathbf {p}_{r}}{\Vert \mathbf {p}_{s} - \mathbf {p}_{r} \Vert })^{T} \Delta _{s} = 0$. This implies that if the perturbations are small, then the deviation on the specular reflection point is orthogonal to the scattering vector. More exactly, let $\mathbf {C}_{n}^{e}$ be the transformation from the local east–north–up coordinates to the ECEF frame; then, the deviation $\Delta _{s}$ can be expressed as $$\begin{equation*} \Delta _{s} = \mathbf {C}_{n}^{e} \left[ \begin{array}{c}\delta _{e} \\ \delta _{n} \\ \delta _{u} \end{array} \right] \end{equation*}$$ where $\delta _{e}$, $\delta _{n}$, and $\delta _{u}$ are errors in the east, north, and up directions, respectively. Suppose that the latitude and longitude of the nominal specular reflection point are $\phi$ and $\lambda$, respectively; then, the deviation $\Delta _{s}$ can be approximated as $$\begin{equation*} \Delta _{s} = \underbrace{\left[ \begin{array}{cc}- \sin \lambda & - \cos \lambda \, \sin \phi \\ \cos \lambda & - \sin \lambda \, \sin \phi \\ 0 & \cos \phi \end{array} \right]}_{\mathbf {P}} \left[ \begin{array}{c}\delta _{e} \\ \delta _{n} \end{array} \right]. \end{equation*}$$ Consequently $$\begin{equation*} \mathbb {E}\left\lbrace \left[ \begin{array}{cc}\delta _{e}^{2} & \delta _{e} \delta _{n} \\ \delta _{e} \delta _{n} & \delta _{n}^{2} \end{array} \right] \right\rbrace = \mathbf {G}_{s}^{T} \, \left[ \begin{array}{cc}\sigma _{t}^{2} \mathbf {I}& 0 \\ 0 & \sigma _{r}^{2} \mathbf {I}\end{array} \right] \, \mathbf {G}_{s} \tag{50} \end{equation*}$$ where $$\begin{equation*} \mathbf {G}_{s} = \left[ \begin{array}{c}\mathbf {A}_{t}^{T} \\ \mathbf {A}_{r}^{T} \end{array} \right] \mathbf {A}_{s} \mathbf {P}(\mathbf {P}^{T} \mathbf {A}_{s}^{T} \mathbf {A}_{s} \mathbf {P})^{-1}. \end{equation*}$$ The error ellipse can then be obtained from the above equation for the assessment of the error in the local horizontal plane.

SECTION IV. Illustrative Simulation Results

In this section, the proposed algorithm for the determination of the specular reflection point is simulated by using the TRITON mission as an example. The TRITON is a small satellite that will operate in the 500-km sun synchronous orbit to perform GNSS-R remote sensing experiment [24]. The onboard GNSS-R payload is capable of processing reflected GPS, Galileo, and QZSS signals [24].

A. Example 1

In the TRITON GNSS-R receiver, the position of a GNSS satellite is obtained from broadcast navigation messages, and position of the TRITON satellite is estimated by the onboard GNSS receiver based on the measurement of the direct line-of-sight GNSS signals. These two position data serve as inputs for the application of the proposed method in the determination of the specular reflection point. In this example, the transmitter and the receiver at one epoch are located at $\mathbf {p}_{t} = \left[ \scriptsize\begin{array}{c}0.53812838 \\ 3.70339643 \\ -1.86697799 \end{array}\right ] \, a$ and $\mathbf {p}_{r} = \left[ \scriptsize\begin{array}{c}-0.81394480 \\ 0.62674404 \\ -0.34731185 \end{array} \right] \, a$, respectively. Here, $a$ is the semimajor axis of the Earth. Following the orthonormalization and decomposition procedure, the two vectors $\mathbf {n}_{1}$ and $\mathbf {n}_{2}$ are given by $\mathbf {n}_{1} = \left[ \scriptsize\begin{array}{c}0.12867282 \\ 0.88552566 \\ -0.44641640 \end{array} \right]$ and $\mathbf {n}_{2} = \left[ \scriptsize\begin{array}{c}-0.99120687 \\ 0.10083492 \\ -0.08568117 \end{array} \right]$, respectively. The matrix $\mathbf {N}$ is then formed, and the three polynomial equations (32), (35), and (36) in terms of $\mathbf {z}$ are obtained. The coefficients in (35) and (36) are given in (51) and (52), respectively. The polynomial equations are then reduced through the resultant technique to obtain a polynomial equation of one variable $z_{1}$, which is given in (53). This polynomial contains six real roots and five pairs of complex roots. The complex roots are discarded, and the real roots are 0.25161743, 0.68938695, $-\text{0.66128242}$, $-\text{0.94007201}$, $-\text{4.39991249} \times \text{10}^{\text{10}}$, and $\text{4.39991213} \times \text{10}^{\text{10}}$. The last two real roots can be neglected as they violate the condition (32). For each legitimate candidate $z_{1}$, one proceeds to solve $z_{2}$ of (38) and (39). Thereafter, the variable $z_{3}$ is determined based on the original equations. Through this back substitution, one obtains four candidate solutions of $\mathbf {z}$ as in (54). The distance of each candidate is examined, and the minimal distance is achieved by the third candidate in (54). Thus, the specular reflection point $\mathbf {p}_{s}$ is $$\begin{equation*} \mathbf {p}_{s} = a \, {\boldsymbol{\Sigma }}\, \left[ \begin{array}{r}-0.66128242 \\ 0.65858232 \\ -0.35913073 \end{array} \right] = \left[ \begin{array}{r}-0.66128242 \\ 0.65858232 \\ -0.35792663 \end{array} \right] \, a. \end{equation*}$$ It can be further verified that this point satisfies (1), (3), and (6).

Note that the leading coefficients of the polynomial in (53) are very small. If we only consider the sixth-order polynomial by neglecting terms of order seven and above, the six-order polynomial yields four real roots and one pair of complex root. The real roots are 0.25161743, 0.68938852, $-\text{0.66128592}$, and $-\text{0.94004563}$, which match the real roots mentioned above at least to the four digits after the decimal point. This suggests that, in practice, it may suffice to solve a sextic polynomial equation [Eq. (51--54) shown at the bottom of the next page] $$\begin{align*} \left[ \begin{array}{c}h_{1} \\ h_{2} \\ h_{3} \end{array} \right] =& \left[ \begin{array}{r}-0.11611616 \\ 1.70651539 \\ 3.36290421 \end{array} \right] \; \; \text{and} \; \; \left[ \begin{array}{c}k_{1} \\ k_{2} \end{array} \right] = \left[ \begin{array}{r}-1.35207318 \\ -3.07665239 \end{array} \right] \tag{51} \\ \left[ \begin{array}{c}\overline{r}_{1} \\ \overline{r}_{2} \\ \overline{r}_{3} \end{array} \right] =& \left[ \begin{array}{r}- 0.81394480 \\ 0.62674404 \\ - 0.34848024 \end{array} \right], \;\\ \left[ \begin{array}{c}f_{1} \\ f_{2} \\ f_{3} \end{array} \right] =& \left[ \begin{array}{r}15.51534427 \\ - 1.57474635 \\ 1.35288451 \end{array} \right], \; \left[ \begin{array}{c}\overline{t}_{1} \\ \overline{t}_{2} \\ \overline{t}_{3} \end{array} \right] = \left[ \begin{array}{r}0.53812838 \\ 3.70339643 \\ -1.87325870 \end{array} \right], \; \text{and} \; \left[ \begin{array}{c}g_{1} \\ g_{2} \\ g_{3} \end{array} \right] = \left[ \begin{array}{r}2.67924423 \\ 2.75010700 \\ -1.32066215 \end{array} \right] \tag{52} \\ p_{1}(z_{1}) = & -4.16088 \times 10^{-42} z_{1}^{16} -2.17078 \times 10^{-38} z_{1}^{15} + 8.05514 \times 10^{-21} z_{1}^{14} + 1.26193 \times 10^{-17} z_{1}^{13} + 9.35488 \times 10^{-15} z_{1}^{12} \\ & + 4.30245 \times 10^{-12} z_{1}^{11} + 1.34223 \times 10^{-9} z_{1}^{10} + 2.92222 \times 10^{-7} z_{1}^{9} + 4.43736 \times 10^{-5} z_{1}^{8} + 0.004601 z_{1}^{7} \\ & + 0.310983 z_{1}^{6} + 12.417158 z_{1}^{5} + 226.27490 z_{1}^{4} + 135.44913 z_{1}^{3} -159.18969 z_{1}^{2} -65.784665 z_{1} + 23.553757 \\ = & 0 \tag{53} \\ &\text{solutions of } \mathbf {z}= \left[ \begin{array}{r}0.25161743 \\ -0.86023112 \\ 0.44349869 \end{array} \right], \left[ \begin{array}{r}0.68938695 \\ 0.65559669 \\ -0.30812110 \end{array} \right], \left[ \begin{array}{r}-0.66128242 \\ 0.65858232 \\ -0.35913073 \end{array} \right], \; \text{and} \; \left[ \begin{array}{r}-0.94007201 \\ 0.28931094 \\ -0.18045443 \end{array} \right] \tag{54} \end{align*}$$

B. Example 2

In the spaceborne GNSS-R receiver, the specular reflection points with respect to admissible GNSS satellites are computed. Fig. 1 depicts a snapshot of the subsatellite points of the TRITON satellite, GPS satellites, and QZSS satellites. Different GNSS satellites are annotated with their PRNs. Circled GPS/QZSS satellites are those can render reflected signals. The specular reflection points with respect to these GPS/QZSS are depicted in terms of square symbols in Fig. 2. It is noted that when the TRITON is in the Asian Pacific region, the number of reflected signals is increased due to QZSS satellites. The GNSS-R processing technique for QZSS signals has been discussed in [24].

Fig. 1.

Specular reflection point computation.

Fig. 2.

Specular reflection points.

The DOP metrics in reflection, as defined in (47) and (48), are then evaluated. It is noted that the $\text{DOPR}_{t}$ and $\text{DOPR}_{r}$ may be related as $\mathbf {A}_{t}$ and $\mathbf {A}_{r}$ in (44) bear a similar form. Indeed, given that both TRITON and GNSS satellites under consideration are near-circular orbiting satellites, the DOP in reflection is related to the incidence angle at the specular reflection point. Fig. 3 depicts the values of $\text{DOPR}_{t}$, $\text{DOPR}_{r}$, and incidence angle of the TRITON satellite with respect to one GPS satellite. Here, the incidence angle is measured from the zenith, and the unit is in radian. It is observed that the three variables are indeed highly correlated. It is also noted that the $\text{DOPR}_{t}$ is much smaller than the $\text{DOPR}_{r}$ due to the fact that the orbit of the GNSS satellite is much higher than that of the TRITON satellite. Thus, for the GNSS-R mission, the analysis of $\text{DOPR}_{r}$ is more relevant. In the operation of the GNSS-R receiver, an elevation mask is typically set in processing the observed signals. As a result, both $\text{DOPR}_{t}$ and $\text{DOPR}_{r}$ are bounded. For the TRITON satellite at 500 km, the value of $\text{DOPR}_{r}$ varies and yet is bounded by 1.6, implying that the error will not be significantly amplified due to orbit determination error. Fig. 4 further depicts the correlation between $\text{DOPR}_{t}$, $\text{DOPR}_{r}$, and incidence angle with respect to all the permissible GPS and QZSS satellites. The density plot in the top-left plate depicts $\text{DOPR}_{r}$ versus $\text{DOPR}_{t}$ for a collection of 1 546 499 samples with respect to GPS satellites. The linear curve that fits the data is also shown, in which the root-mean-square error is 0.0222 and the $R^{2}$ value is 0.956. The top-right plate reveals the correlation between the $\text{DOPR}_{r}$ and the incidence angle for the same set of data, leading to the root-mean-square error of 0.0257 and the $R^{2}$ value of 0.942. The bottom two plates are corresponding results with respect to QZSS satellites. It is observed that the $\text{DOPR}_{t}$ of the QZSS satellite is smaller due to the factor $\frac{1}{\Vert \mathbf {p}_{s} - \mathbf {p}_{t}\Vert }$ in the matrix $\mathbf {A}_{r}$ and the influence on the $\text{DOPR}_{t}$. The number of samples with respect to QZSS satellites is 290 267. The $R^{2}$ values of the linear models are 0.896 and 0.948, respectively. Thus, one can approximate the relationship between $\text{DOPR}_{t}$, $\text{DOPR}_{r}$, and incidence angle through a linear model.

Fig. 3.

DOP in reflection.

Fig. 4.

Correlation between $\text{DOPR}_{t}$, $\text{DOPR}_{r}$, and incidence angle.

The errors in the horizontal plane are finally assessed. Simulation results of the relative ground tracks of the TRITON satellite, the GNSS satellite, and the reflection point are depicted in Fig. 5, in which the TRITON satellite is undergone a south to north movement. During this period, the ground track of the specular reflection lies between the ground tracks of the satellites. Note that the ground track of the specular reflection point can be at high latitude, implying that the potential of TRITON GNSS-R mission in ice layer or cryosphere research. With the analysis of the dilution of procession in reflection, it is possible to assess the errors of the specular reflection point. Assume that $\sigma _{t} = 1$ m and $\sigma _{r} = 2$ m; the error ellipses of the east–north errors can be computed from (50). In addition to showing the tracks of the reflection point with respect to a GNSS satellite, Fig. 5 in some separate boxes depicts the 95% error ellipses in the horizontal plane at some points in which the units in the error are in meters. It is observed that the orientation of the error ellipses varies as the reflection geometry changes. The results can be used in the quantification of the GNSS-R remote sensing data.

Fig. 5.

Error analysis.

SECTION V. Conclusion

This article proposes a method for the determination of the specular reflection point for GNSS-R mission. An analytic approach is developed to characterize the specular reflection point on the ellipsoidal Earth. Moreover, it is shown that the specular reflection point is characterized by a polynomial equation of generic degree 16, leading to a computational approach that is prone to neither iterations nor uncertainties on the initial estimate. A sensitivity analysis is then conducted, and two DOP metrics in reflection are introduced. The proposed procedure and analysis approaches are verified by using the TRITON orbit information with respect to GPS/QZSS satellites. The procedure is implemented in the TRITON GNSS-R payload, and the analysis results can be used for the control of payload operation as well as GNSS-R remote sensing data processing.

This article has investigated the reflection point on the ellipsoidal Earth. However, the solving of a 16-degree polynomial remains challenging in real-time operation. As the eccentricity of the Earth is close to zero, it may be possible to derive a low-degree polynomial approximation, as discussed in Example 1. This subject will be investigated in the future. In practice, the actual reflection point depends on the terrain characteristics. It is believed that the proposed approach can pave the way for the inclusion of local terrain models. This subject will also be further researched.