# Nested Coordinate Systems in Geometric Algebra

Garret Sobczyk
Universidad de las Américas-Puebla
Departamento de Actuaría Física y Matemáticas
72820 Puebla, Pue., México

###### Abstract

A nested coordinate system is a reassigning of independent variables to take
advantage of geometric or symmetry properties of a particular application.
Polar, cylindrical and spherical coordinate systems are primary examples of
such a regrouping that have proved their importance in the separation of
variables method for solving partial differential equations. Geometric algebra
offers powerful complimentary algebraic tools that are unavailable in other
treatments.

AMS Subject Classification MSC-2020: 15A63, 15A67, 42B37.

Keywords: Clifford algebra, coordinate systems, geometric algebra, separation
of variables.

## 0 Introduction

Geometric algebra $\mathbb{G}_{3}$ is the natural generalization of Gibbs
Heaviside vector algebra, but unlike the latter, it can be immediately
generalized to higher dimensional geometric algebras $\mathbb{G}_{p,q}$ of a
quadratic form. On the other hand, Clifford analysis, the generalization of
Hamilton’s quaternions, is also expressed in Clifford’s geometric algebras
[1]. The main purpose of this article is to formulate the concept of a nested
coordinate system, a generalization of the well-known methods of orthogonal
coordinate systems to apply to any coordinate system. We restrict ourselves to
the geometric algebra $\mathbb{G}_{3}$ because of its close relationship to
the Gibbs-Heaviside vector calculus [3]. This restriction also draws attention
to the clear advantages of geometric algebra over the later, because of its
powerful associative algebraic structure.

The idea of a nested rectangular coordinate system arises naturally when
studying properties of polar coordinates in the $2$ and $3$-dimensional
Euclidean vector spaces $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$. We begin by
discussing the relationship between ordinary polar coordinates and the nested
rectangular coordinate system ${\cal N}_{1,2}$, before going on to the higher
dimensional nested coordinate system ${\cal N}_{1,2,3}$ utilized in the
reformulation of cylindrical and spherical coordinates. A detailed discussion
of the geometric algebra $\mathbb{G}_{3}$ is not given here, but results are
often expressed in the closely related well-known Gibbs-Heaviside vector
analysis for the benefit of the reader.

##  1 Polar and nested coordinates systems

Let $\mathbb{G}_{2}:=\mathbb{G}(\mathbb{R}^{2})$ be the geometric algebra of
$2$-dimensional Euclidean space $\mathbb{R}^{2}$. An introductory treatment of
the geometric algebras $\mathbb{G}_{1}$, $\mathbb{G}_{2}$ and $\mathbb{G}_{3}$
is given in [4, 5, 6]. Most important in studying the geometry of the
Euclidean plane is the position vector

$\mathbf{x}:=\mathbf{x}[x,\hat{x}]=x\hat{x}$ (1)

expressed here as a product of its Euclidean magnitude $x$ and its unit
direction, the unit vector $\hat{x}$. In terms of rectangular coordinates
$(x_{1},x_{2})\in\mathbb{R}^{2}$,

$\mathbf{x}=\mathbf{x}[x_{1},x_{2}]=x_{1}e_{1}+x_{2}e_{2},$ (2)

for the orthogonal unit vectors $e_{1},e_{2}$ along the $x_{1}$ and $x_{2}$
axis, respectively. The advantage of our notation is that it immediately
generalizes to $3$ and higher dimensional spaces of arbitrary signature
$(p,q)$ in any of the definite geometric algebras
$\mathbb{G}_{p,q}:=\mathbb{G}(\mathbb{R}^{p,q})$ of a quadratic form.

The vector derivative, or gradient in the Euclidean plane is defined by

$\nabla:=e_{1}\partial_{1}+e_{2}\partial_{2}$ (3)

where $\partial_{1}:=\frac{\partial}{\partial x_{1}}$ and
$\partial_{2}:=\frac{\partial}{\partial x_{2}}$ are partial derivatives [3,
p.105]. Clearly,

$e_{1}=\partial_{1}\mathbf{x}=e_{1}\cdot\nabla\mathbf{x},\ \
e_{2}=\partial_{2}\mathbf{x}=e_{2}\cdot\nabla\mathbf{x}.$

Since $\nabla$ is the usual 2-dimensional gradient, it has the well-known
properties

$\nabla\mathbf{x}=2,\quad{\rm and}\quad\nabla x=\hat{x}.$

With the help of the product rule for differentiation,

$2=\nabla\mathbf{x}=(\nabla
x)\hat{x}+x(\nabla\hat{x})=\hat{x}^{2}+x(\nabla\hat{x}).$ (4)

Since in geometric algebra $\mathbf{x}^{2}=x^{2}$, it follows that
$\hat{x}^{2}=1$, so that for $\mathbf{x}\in\mathbb{R}^{2}$,

$\nabla\hat{x}=\frac{1}{x}\ \ {\rm and}\ \ e_{1}\cdot\nabla
x=e_{1}\cdot\hat{x}=\frac{x_{1}}{x},\ \ e_{2}\cdot\nabla
x=e_{2}\cdot\hat{x}=\frac{x_{2}}{x}.$ (5)

Similarly, $\nabla\hat{x}=\frac{n-1}{x}$ for $\mathbf{x}\in\mathbb{R}^{n}$.
This is the first of many demonstrations of the power of geometric algebra
over standard vector algebra.

By a nested rectangular coordinate system ${\cal
N}_{1,2}(x_{1},x[x_{1},x_{2}])$, we mean

$\mathbf{x}=x\hat{x}=\mathbf{x}[x_{1},x]=\mathbf{x}\big{[}x_{1},x[x_{1},x_{2}]\big{]}.$

The grouping of the variables allows us to consider $x_{1}$ and
$x:=\sqrt{x_{1}^{2}+x_{2}^{2}}$ to be independent. The partial derivatives
with respect to these independent variables is denoted by
$\hat{\partial}_{1}:=\frac{\hat{\partial}}{\partial x_{1}}$ and
$\hat{\partial}_{x}:=\frac{\hat{\partial}}{\partial x}$, the hat on the
partial derivatives indicating the new choice of independent variables.

For polar coordinates $(x,\theta)\in\mathbb{R}^{2}$, for
$x:=\sqrt{x_{1}^{2}+x_{2}^{2}}\geq 0$, $0\leq\theta<2\pi$, and
$\mathbf{x}:=\mathbf{x}[x,\theta]$,

$\mathbf{x}=x\hat{x}[\theta]=x(e_{1}\frac{x_{1}}{x}+e_{2}\frac{x_{2}}{x})=x(e_{1}\cos\theta+e_{2}\sin\theta),$
(6)

where $\cos\theta:=\frac{x_{1}}{x}$ and $\sin\theta:=\frac{x_{2}}{x}$. Using
(5),

$\nabla\hat{x}=\nabla\hat{x}[\theta]=(\nabla\theta)\frac{\partial\hat{x}}{\partial\theta}=\frac{1}{x}\quad\iff\quad\nabla\theta=\frac{1}{x}\frac{\partial\hat{x}}{\partial\theta},\
\nabla^{2}\theta=0,$ (7)

since

$\nabla\hat{x}=(\nabla\theta)\partial_{\theta}(e_{1}\cos\theta+e_{2}\sin\theta)=(\nabla\theta)(-e_{1}\sin\theta+e_{2}\cos\theta),$

and

$\nabla^{2}\theta=-\frac{\hat{x}}{x^{2}}\partial_{\theta}\hat{x}+\frac{1}{x}(\nabla\theta)\partial_{\theta}^{2}\hat{x}=-2\Big{(}\frac{\hat{x}}{x^{2}}\cdot(\partial_{\theta}\hat{x})\Big{)}=0.$

The $\iff$ follows by multipling both sides of the first equation by the unit
vector $\partial_{\theta}\hat{x}$, which is allowable in geometric algebra.
Note also the use of the famous geometric algebra identity
$2\mathbf{a}\cdot\mathbf{b}=(\mathbf{a}\mathbf{b}+\mathbf{b}\mathbf{a})$ for
vectors $\mathbf{a}$ and $\mathbf{b}$, [4, p.26].

The 2-dimensional gradient $\nabla$,

$\nabla=e_{1}\frac{\partial}{\partial x_{1}}+e_{2}\frac{\partial}{\partial
x_{2}}=e_{1}\partial_{1}+e_{2}\partial_{2}$ (8)

already defined in (3), and the Laplacian $\nabla^{2}$ is given by

$\nabla^{2}=\frac{\partial^{2}}{\partial
x_{1}^{2}}+\frac{\partial^{2}}{\partial
x_{2}^{2}}=\partial_{1}^{2}+\partial_{2}^{2}.$ (9)

In polar coordinates,

$\hat{\nabla}=(\nabla
x)\hat{\partial}_{x}+(\nabla\theta)\hat{\partial}_{\theta}=\hat{x}\,\hat{\partial}_{x}+\frac{1}{x}(\hat{\partial}_{\theta}x)\hat{\partial}_{\theta}$
(10)

for the gradient where
$\hat{\partial}_{\theta}:=\frac{\hat{\partial}}{\partial\theta}$, and since
$\hat{\nabla}^{2}\theta=0$,

$\hat{\nabla}^{2}=\hat{\nabla}\big{(}\hat{x}\,\hat{\partial}_{x}+(\hat{\nabla}\theta)\,\hat{\partial}_{\theta}\big{)}=\Big{(}\hat{\nabla}\hat{x}+\hat{x}\cdot\hat{\nabla}\Big{)}\hat{\partial}_{x}+\Big{(}\hat{\nabla}^{2}\theta+(\hat{\nabla}\theta)\cdot\hat{\nabla}\Big{)}\hat{\partial}_{\theta}$
$=\hat{\partial}_{x}^{2}+\frac{1}{x}\hat{\partial}_{x}+\frac{1}{x^{2}}\hat{\partial}_{\theta}^{2}.$
(11)

for the Laplacian. The decomposition of the Laplacian (11), directly implies
that Laplace’s differential equation is separable in polar coordinates.

When expressed in nested rectangular coordinates ${\cal N}_{1,2}(x_{1},x)$,
the gradient $\nabla\equiv\hat{\nabla}$ takes the form

$\hat{\nabla}:=(\nabla x_{1})\frac{\hat{\partial}}{\partial x_{1}}+(\nabla
x)\frac{\hat{\partial}}{\partial
x}=e_{1}\hat{\partial}_{1}+\hat{x}\hat{\partial}_{x}.$ (12)

Dotting equations (8) and (12) on the left by $e_{1}$ and $\hat{x}$ gives the
transformation rules

$\partial_{1}=\hat{\partial}_{1}+\frac{x_{1}}{x}\hat{\partial}_{x},\ \
\hat{x}\cdot\hat{\nabla}=\frac{x_{1}}{x}\hat{\partial}_{1}+\hat{\partial}_{x}=\cos\theta\,\partial_{1}+\sin\theta\,\partial_{2}.$

Using these formulas the nested Laplacian takes the form

$\hat{\nabla}^{2}=\hat{\partial}_{1}^{2}+2\frac{x_{1}}{x}\hat{\partial}_{x}\hat{\partial}_{1}+\frac{1}{x}\hat{\partial}_{x}+\hat{\partial}_{x}^{2}=-\hat{\partial}_{1}^{2}+2\partial_{1}\hat{\partial}_{1}+\frac{1}{x}\hat{\partial}_{x}+\hat{\partial}_{x}^{2}.$
(13)

The unusual feature of the nested Laplacian is that it is defined in terms of
both the ordinary partial derivative $\partial_{1}$ and the nested partial
derivative $\hat{\partial}_{1}$. Whereas partial derivatives generally
commute, partial derivatives of different types do not. For example, it is
easily verified that

$\partial_{1}\hat{\partial}_{1}x_{1}x^{2}=2x_{1},\ \ {\rm whereas}\ \
\hat{\partial}_{1}\partial_{1}x_{1}x^{2}=4x_{1}.$

Because the mixed partial derivatives $\hat{\partial}_{x}\hat{\partial}_{1}$
occurs in (13), Laplace’s differential equation in the real rectangular
coordinate system ${\cal N}_{1,2}(x_{1},x)$ is not, in general, separable.
Indeed, suppose that a harmonic function $F$ is separable, so that $F=X_{1}X$
for $X_{1}=X_{1}[x_{1}],X=X[x]$. Using (13),

$\frac{\hat{\nabla}^{2}F}{X_{1}X}=\frac{\partial_{1}^{2}X_{1}}{X_{1}}+\frac{\Big{(}\partial_{x}^{2}X+\frac{1}{x}\partial_{x}\Big{)}X}{X}+2\Big{(}\frac{x_{1}\partial_{1}X_{1}}{X_{1}}\Big{)}\Big{(}\frac{\partial_{x}X}{xX}\Big{)}=0.$
(14)

The last term on the prevents F in general from being separable. However, it
is easily checked that $F=k\frac{x_{1}}{x^{2}}$ is harmonic and a solution of
(13). When $X_{1}[x_{1}]=kx_{1}$, it is easily checked that
$\frac{x_{1}\partial_{1}X_{1}}{X_{1}}=1$. Letting $F=kx_{1}X[x]$, and
requiring $\hat{\nabla}^{2}F=0$, leads to the differential equation for
$X[x]$,

$3\partial_{x}X+x\partial_{x}^{2}X=0,$

with the solution $X[x]=c_{1}\frac{1}{x^{2}}+c_{2}$. The simplest example of a
harmonic function $F=X_{1}X$ is when $X_{1}=x_{1}$ and $X=\frac{1}{x^{2}}$. A
graph of this function is shown in Figure 1.

Figure 1: The harmonic $2$-dimensional function
$F=\frac{x_{1}}{x_{1}^{2}+x_{2}^{2}}$ is shown.

##  2 Special harmonic functions in nested coordinates

Consider the real nested rectangular coordinate system $(x_{1},x_{p},x)$,
defined by

${\cal N}_{1,2,3}:=\\{(x_{1},x_{p},x)|\
\mathbf{x}=x\hat{x}=x_{1}e_{1}+x_{p}\hat{x}_{p}+x\hat{x}\\},$

where $x_{p}=\sqrt{x_{1}^{2}+x_{2}^{2}}\geq 0,\
x=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}\geq 0$. In nested coordinates, the
gradient $\nabla=e_{1}\partial_{1}+e_{2}\partial_{2}+e_{3}\partial_{3}$ takes
the form

$\hat{\nabla}=(\hat{\nabla}x_{1})\hat{\partial}_{1}+(\hat{\nabla}x_{p})\hat{\partial}_{p}+(\hat{\nabla}x)\hat{\partial}_{x}=e_{1}\hat{\partial}_{1}+\hat{x}_{p}\hat{\partial}_{p}+\hat{x}\hat{\partial}_{x},$
(15)

where $\hat{\partial}_{p}:=\frac{\hat{\partial}}{\partial x_{p}}$. Formulas
relating the gradients $\nabla$ and $\hat{\nabla}$ easily follow:

$\partial_{1}=\hat{\partial}_{1}+e_{1}\cdot\hat{x}_{p}\,\hat{\partial}_{p}+e_{1}\cdot\,\hat{x}\,\hat{\partial}_{x}=\hat{\partial}_{1}+\frac{x_{1}}{x_{p}}\,\hat{\partial}_{p}+\frac{x_{1}}{x}\,\hat{\partial}_{x}$
(16)
$\partial_{2}=e_{2}\cdot\hat{x}_{p}\,\hat{\partial}_{p}+e_{2}\cdot\,\hat{x}\,\hat{\partial}_{x}=\frac{x_{2}}{x_{p}}\,\hat{\partial}_{p}+\frac{x_{2}}{x}\,\hat{\partial}_{x}$
(17)

and

$\partial_{3}=e_{3}\cdot\,\hat{x}\,\hat{\partial}_{x}=\frac{x_{3}}{x}\,\hat{\partial}_{x}.$
(18)

For the Laplacian $\nabla^{2}$ in nested coordinates, with the help of (15),

$\hat{\nabla}^{2}=\hat{\nabla}(e_{1}\hat{\partial}_{1}+\hat{x}_{p}\partial_{x_{p}}+\hat{x}\hat{\partial}_{x})=e_{1}\cdot\hat{\nabla}\,\hat{\partial}_{1}+\hat{\nabla}\cdot\hat{x}_{p}\,\hat{\partial}_{p}+\hat{\nabla}\cdot\hat{x}\,\hat{\partial}_{x}$
$=e_{1}\cdot\hat{\nabla}\,\hat{\partial}_{1}+\hat{\nabla}\cdot\hat{x}_{p}\,\hat{\partial}_{p}+\hat{\nabla}\cdot\hat{x}\,\hat{\partial}_{x}$

$=\Big{(}\hat{\partial}_{1}+\frac{x_{1}}{x_{p}}\,\hat{\partial}_{p}+\frac{x_{1}}{x}\,\hat{\partial}_{x}\Big{)}\hat{\partial}_{1}+\Big{(}\frac{1}{x_{p}}+\frac{x_{1}}{x_{p}}\hat{\partial}_{1}+\hat{\partial}_{p}+\frac{x_{p}}{x}\hat{\partial}_{p}\Big{)}\hat{\partial}_{p}$
$+\Big{(}\frac{2}{x}+\frac{x_{1}}{x}\hat{\partial}_{1}+\frac{x_{p}}{x}\hat{\partial}_{p}+\hat{\partial}_{x}\Big{)}\hat{\partial}_{x}$
$=\hat{\partial}_{1}^{2}+\hat{\partial}_{p}^{2}+\hat{\partial}_{x}^{2}+2\bigg{(}\frac{x_{1}}{x_{p}}\hat{\partial}_{1}\hat{\partial}_{p}+\frac{x_{1}}{x}\hat{\partial}_{1}\hat{\partial}_{x}+\frac{x_{p}}{x}\hat{\partial}_{p}\hat{\partial}_{x}\bigg{)}+\frac{1}{x_{p}}\hat{\partial}_{p}+\frac{2}{x}\hat{\partial}_{x}.$
(19)

Another expression for the Laplacian in mixed coordinates is obtained with the
help of (16),

$\hat{\nabla}^{2}=-\hat{\partial}_{1}^{2}+\hat{\partial}_{p}^{2}+\hat{\partial}_{x}^{2}+2\bigg{(}\partial_{1}\hat{\partial}_{1}+\frac{x_{p}}{x}\hat{\partial}_{p}\hat{\partial}_{x}\bigg{)}+\frac{1}{x_{p}}\hat{\partial}_{p}+\frac{2}{x}\hat{\partial}_{x}.$
(20)

Suppose $F=F[x_{1},x_{p},x]$. In order for $F$ to be harmonic,
$\hat{\nabla}^{2}F=0$. Assuming that $F$ is separable,
$F=X_{1}[x_{1}]X_{p}[x_{p}]X_{x}[x]$, and applying the Laplacian (20) to $F$
gives

$\hat{\nabla}^{2}F=(\hat{\partial}_{1}^{2}X_{1})X_{p}X_{x}+X_{1}\Big{(}\big{(}\hat{\partial}_{p}^{2}+\frac{1}{x_{p}}\hat{\partial}_{p}\big{)}X_{p}\Big{)}X_{x}+X_{1}X_{p}\Big{(}\frac{2}{x}\hat{\partial}_{x}X_{x}\Big{)}$
$+2\bigg{(}\big{(}x_{p}\partial_{p}X_{p}\big{)}\big{(}\frac{1}{x}\hat{\partial}_{x}X_{x}\big{)}X_{1}+\big{(}\partial_{1}X_{p}\big{)}X_{x}+X_{p}\big{(}\partial_{1}X_{x}\big{)}\bigg{)}.$
(21)

We now calculate the interesting expression

$\frac{\big{(}x_{p}\partial_{p}X_{p}\big{)}\big{(}\frac{1}{x}\hat{\partial}_{x}X_{x}\big{)}X_{1}+\big{(}\partial_{1}X_{p}\big{)}X_{x}+X_{p}\big{(}\partial_{1}X_{x}\big{)}}{X_{1}X_{p}X_{x}}$
$=\Big{(}x_{p}\big{(}\partial_{p}\log
X_{p}\big{)}\Big{)}\Big{(}\big{(}\frac{1}{x}\partial_{x}\log
X_{x}\big{)}\Big{)}+\frac{\partial_{1}\log(X_{p}X_{x})}{X_{1}}.$

In general, because of the last term in (21), a function $F=X_{1}X_{p}X_{x}$
will not be separable. However, just as in the two dimensional case, there are
$3$-dimensional harmonic solutions of the form $F=x_{1}^{k}x_{p}^{m}x^{n}$.
Taking the Laplacian (19) of $F$, with the help of [7], gives

$\hat{\nabla}^{2}F=(2km+m^{2})x^{n}x_{1}^{k}x_{p}^{m-2}+(-k+k^{2})x^{n}x_{1}^{k-2}x_{p}^{m}$
$+(2kn+2mn+n(1+n))x^{n-2}x_{1}^{k}x_{p}^{m}=0.$

This last expression vanishes when the system of three equations,

$\\{2km+m^{2}=0,\ \ -k+k^{2}=0,\ \ {\rm and}\ \ 2kn+2mn+n(1+n)=0\\}.$

All of the distinct non-trivial harmonic solutions $F=x_{1}^{k}x_{p}^{m}x^{n}$
are listed in the following Table

k | m | n
---|---|---
1 | 0 | 0
0 | 0 | -1
1 | -2 | 0
1 | 0 | -3
1 | -2 | 1
(22)

##  3 Cylindrical and spherical coordinates

Cylindrical and spherical coordinates are examples of nested coordinates
${\cal N}_{1,2}(\mathbb{R})$, and ${\cal N}_{2,3}(\mathbb{R})$, respectively.
For the first,

$\mathbf{x}=\mathbf{x}[x_{p},\theta,x_{3}]=\mathbf{x}_{p}[x_{p},\theta]+\mathbf{x}_{3}[x_{3}],$
(23)

where $\mathbf{x}_{p}=x_{p}\hat{x}_{p}[\theta]$,
$x_{p}=\sqrt{x_{1}^{2}+x_{2}^{2}}$, and $\mathbf{x}_{3}=x_{3}e_{3}$.
Cylindrical coordinates
$(x_{p},\theta,x_{3})\in\mathbb{R}^{3}=\mathbb{R}^{2}\cup\mathbb{R}^{1}$ are a
decomposition of $\mathbb{R}^{3}$ into the polar coordinates
$(x_{p},\theta)\in\mathbb{R}^{2}$, already studied in Section 1, and
$x_{3}\in\mathbb{R}^{1}$. For spherical coordinates,
$\mathbf{x}_{p}=x_{p}\hat{x}_{p}[\theta]$ the same as in cylindrical and polar
coordinates, and

$\mathbf{x}=\mathbf{x}[x,\theta,\varphi]=x\hat{x}[\theta,\varphi])=x\Big{(}e_{3}\cos\varphi+\hat{x}_{p}[\theta]\sin\varphi\Big{)},$
(24)

where

$x=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}},\
\hat{x}[\theta,\varphi]=e_{3}\cos\varphi+\hat{x}_{p}[\theta]\sin\varphi,\
\hat{x}_{p}[\theta]=e_{1}\cos\theta+e_{2}\sin\theta.$

The basic quantities that define both cylindrical and spherical coordinates
are shown in Figure 2.

Figure 2: For cylindrical coordinates,
$\mathbf{x}=x_{p}\hat{x}_{p}[\theta]+x_{3}e_{3}$. For spherical coordinates,
$\mathbf{x}=x(e_{3}\cos\varphi+\hat{x}_{p}[\theta]\sin\varphi)$.

The gradient $\hat{\nabla}$ and Laplacian $\hat{\nabla}^{2}$ for cylindrical
coordinates are easily calculated. With the help of (7), (10), and (11),

$\hat{\nabla}=(\hat{\nabla}x_{p})\hat{\partial}_{p}+(\hat{\nabla}\theta)\hat{\partial}_{\theta}+(\hat{\nabla}x_{3})\hat{\partial}_{3}$

for the cylindrical gradient, and

$\hat{\nabla}^{2}=\hat{\nabla}\Big{(}\hat{x}_{p}\,\hat{\partial}_{p}+(\hat{\nabla}\theta)\,\hat{\partial}_{\theta}+e_{3}\hat{\partial}_{3}\Big{)}=\hat{\partial}_{p}^{2}+\frac{1}{x_{p}}\hat{\partial}_{p}+\frac{1}{x_{p}^{2}}\hat{\partial}_{\theta}^{2}+\hat{\partial}_{3}^{2}$
(25)

for the cylindrical Laplacian. Letting
$F[\mathbf{x}]=X_{p}[x]X_{\theta}[\theta]X_{3}[x_{3}]$, the resulting equation
is easily separated and solved by standard methods, resulting in three second
order differential equations with solutions,

$X_{p}[x_{p}]=k_{1}J_{n}[\beta x_{p}]+k_{2}Y_{n}[\beta x_{p},]$
$X_{\theta}[\theta]=k_{3}\cos n\theta+k_{4}\sin n\theta$
$X_{3}[x_{3}]=k_{5}\cosh(\alpha(m-x_{3}))+k_{6}\sinh(\alpha(m-x_{3})),$

where $J_{n}$ and $Y_{n}$ are Bessel functions of the first and second kind .
The constants are determined by the various boundary conditions that must be
satisfied in different applications [8, p.254].

Turning to spherical coordinates $(x,\theta,\varphi)\in\mathbb{R}^{3}$, the
spherical gradient

$\hat{\nabla}=(\hat{\nabla}x)\hat{\partial}_{x}+(\hat{\nabla}\theta)\hat{\partial}_{\theta}+(\hat{\nabla}\varphi)\hat{\partial}_{\varphi}=\hat{x}\hat{\partial}_{x}+\frac{1}{x_{p}}(\hat{\partial}_{\theta}\hat{x}_{p})\hat{\partial}_{\theta}+\frac{1}{x}(\hat{\partial}_{\varphi}\hat{x})\hat{\partial}_{\varphi},$
(26)

where from previous calculations for polar and cylindrical coordinates,

$\hat{\nabla}\theta=\frac{1}{x_{p}}(\hat{\partial}_{p}\hat{x}_{p}),\
(\hat{\nabla}\theta)^{2}=\frac{1}{x_{p}^{2}},\ \ \hat{\nabla}^{2}\theta=0,\ \
\hat{\nabla}\varphi=\frac{1}{x}\hat{\partial}_{\varphi}\hat{x},\ \
(\hat{\nabla}\varphi)^{2}=\frac{1}{x^{2}}.$ (27)

Furthermore, since
$\hat{x}=\hat{x}[\theta,\varphi]=e_{3}\cos\varphi+\hat{x}_{p}[\theta]\sin\varphi$

$\frac{2}{x}=\hat{\nabla}\hat{x}=(\hat{\nabla}\theta)(\hat{\partial}_{\theta}\hat{x})+(\hat{\nabla}\varphi)(\hat{\partial}_{\varphi}\hat{x})=\frac{1}{x_{p}}(\hat{\partial}_{\theta}\hat{x})(\hat{\partial}_{\varphi}\hat{x})+\frac{1}{x},$

it follows that

$(\hat{\partial}_{\theta}\hat{x})(\hat{\partial}_{\varphi}\hat{x})=\frac{x_{p}}{x}=\sin\varphi,\
\ {\rm and}\ \ \hat{\nabla}^{2}\varphi=\frac{x_{3}}{x^{2}x_{p}}.$

That $\hat{\nabla}^{2}\varphi=\frac{x_{3}}{x^{2}x_{p}}$ follows using (26) and
(27),

$\hat{\nabla}^{2}\varphi=\hat{\nabla}\big{(}\frac{1}{x}\hat{\partial}_{\varphi}\hat{x}\big{)}=\Big{(}-\frac{\hat{x}}{x^{2}}+\frac{1}{x}\hat{\nabla}\Big{)}\hat{\partial}_{\varphi}\hat{x}$
$=-\frac{\hat{x}}{x^{2}}\hat{\partial}_{\varphi}\hat{x}-\frac{1}{x}\Big{(}(\hat{\nabla}x)\hat{\partial}_{x}\hat{\partial}_{\varphi}\hat{x}+(\hat{\nabla}\theta)\hat{\partial}_{\theta}\hat{\partial}_{\varphi}\hat{x}+(\hat{\nabla}\varphi)\hat{\partial}_{\varphi}^{2}\hat{x}\Big{)}$
$=-\Big{(}\frac{\hat{x}}{x^{2}}\hat{\partial}_{\varphi}\hat{x}+\frac{1}{x^{2}}(\hat{\partial}_{\varphi}\hat{x})\hat{x}\Big{)}+\frac{1}{xx_{p}}(\hat{\partial}_{\theta}\hat{x}_{p})(\hat{\partial}_{\varphi}\hat{\partial}_{\theta}x)=\frac{x_{3}}{x^{2}x_{p}},$

since partial derivatives commute, $\hat{\partial}_{x}\hat{x}=0$, and
$\hat{\partial}_{\varphi}^{2}\hat{x}=-\hat{x}$.

For the spherical Laplacian, using (26) and (27),

$\hat{\nabla}^{2}=\hat{\nabla}\Big{(}\hat{x}\hat{\partial}_{x}+(\hat{\nabla}\theta)\hat{\partial}_{\theta}+(\hat{\nabla}\varphi)\hat{\partial}_{\varphi}\Big{)}$
$=\Big{(}\frac{2}{x}+\hat{\partial}_{x}\Big{)}\hat{\partial}_{x}+(\hat{\nabla}\theta)\cdot\hat{\nabla}\hat{\partial}_{\theta}+\Big{(}\hat{\nabla}^{2}\varphi+(\hat{\nabla}\varphi)\cdot\hat{\nabla}\Big{)}\hat{\partial}_{\varphi}$
$=\Big{(}\hat{\partial}_{x}+\frac{2}{x}\Big{)}\hat{\partial}_{x}+\frac{1}{x_{p}^{2}}\hat{\partial}_{\theta}^{2}+\Big{(}\frac{x_{3}}{x^{2}x_{p}}+\frac{1}{x^{2}}\hat{\partial}_{\varphi}\Big{)}\hat{\partial}_{\varphi},$

equivalent to the usual expression for the Laplacian in spherical coordinates
[8, p.256].

Just as in cylindrical coordinates, the solution of Laplace’s equation in
spherical coordinates is separable,
$F=X_{x}[x]X_{\theta}[\theta]X_{\varphi}[\varphi]$, resulting in three second
order differential equations with solutions

$X_{x}[x]=k_{1}x^{\beta}+k_{2}x^{-(\beta+1)},$ $X_{\theta}[\theta]=k_{3}\cos
n\theta+k_{4}\sin n\theta,$
$X_{\varphi}[\varphi]=k_{5}P_{n}^{m}(\cos\varphi)+k_{6}Q_{n}^{m}(\cos\varphi),$

where $P_{n}^{m}$ and $Q_{n}^{m}$ are the Legendre functions of the first and
second kind, respectively [8, p.258].

## Acknowledgment

This work was largely inspired by a current project that author has with
Professor Joao Morais of Instituto Tecnológico Autónomo de México, utilizing
spheroidal coordinate systems. The struggle with this orthogonal coordinate
system [2], led the author to re-examine the foundations of general coordinate
systems in geometric algebra [5, p.63].

## References

  * [1] R. Ablamowicz, G. Sobczyk, Editors: Lectures on Clifford (Geometric) Algebras and Applications, Birkhäuser, Boston 2003.
  * [2] E. Hobson, 1931, The theory of spherical and ellipsoidal harmonics, Cambridge.
  * [3] J.E. Marsden, A.J. Tromba, Vector Calculus 2nd Ed., Freeman and Company, San Francisco 1980.
  * [4] G. Sobczyk, Matrix Gateway to Geometric Algebra, Spacetime and Spinors, Independent Publisher November 2019. https://www.garretstar.com
  * [5] G. Sobczyk, New Foundations in Mathematics: The Geometric Concept of Number,  Birkhäuser, New York 2013.
  * [6] G. Sobczyk. Many early versions of my work can be found on arXiv, or on my website: https://www.garretstar.com
  * [7] S. Wolfram, Mathematica.
  * [8] Tyn Myint-U, Partial Differential Equations of Mathematical Physics 2nd Ed., North Holland, NY 1980.


# Systematic Study of Ionospheric Scintillation over the Indian Low-Latitudes
during Low Solar Activity conditions

Deepthi Ayyagari<EMAIL_ADDRESS>Abhirup Datta
<EMAIL_ADDRESS>Sumanjit Chakraborty<EMAIL_ADDRESS>Department
of Astronomy, Astrophysics and Space Engineering, Indian Institute of
Technology Indore, Simrol , Indore, 453552, Madhya Pradesh, India  Space and
Atmospheric Sciences Division, Physical Research Laboratory, Ahmedabad 380009,
Gujarat, India.

###### Abstract

A systematic study of ionospheric scintillation at the low-latitudes,
especially around the Equatorial Ionization Anomaly (EIA) and the magnetic
equator, is essential in understanding the dynamics of ionospheric variation
and related physical processes. Our study involves NavIC $S_{4_{C}}$
observations over Indore and Hyderabad. Additionally, GPS $S_{4_{C}}$
observations over Indore were analyzed, under disturbed as well as quiet time
ionospheric conditions from September 2017 through 2019, falling in the
declining phase of the solar cycle 24. The $S_{4_{C}}$ observations were
further analyzed using proxy parameters: ROT and ROTI. These results have been
obtained from three satellites of the NavIC constellation (PRNs 2, 5, and 6).
The onset times of scintillations were observed to be around 19:30 LT (h) and
20:30 LT (h) for Hyderabad and Indore respectively, while the $S_{4_{C}}$ peak
values occurred between 22:00 LT (h) and 23:00 LT (h). The reliability of
NavIC was evaluated using scattering coefficients that revealed a good
correlation across the pair of signals during quiet time ionospheric
conditions. The observations clearly show that the amplitude scintillation of
the NavIC signal follows the Nakagami-m distribution along with the
$\alpha-\mu$ distribution as a depiction of the deep power fades caused by
scintillation on these signals. This paper shows the impact of such systematic
studies near these locations for the first time, in improving the
understanding of the dynamic nature of low-latitude ionosphere under low solar
activity conditions.

###### keywords:

NavIC; GPS; Amplitude Scintillation; Nakagami-m

††journal: Advances in Space Research

##  1 Introduction

The Earth’s ionosphere, which is approximately stratified between 60 and 1000
km above ground, often produces electron density disturbances, which disperse
radio waves passing through it, resulting in amplitude and phase
scintillations. The degree to which the ionospheric irregularities cause
scintillations is determined by the frequency of the signal and the intensity
of the electron density irregularities. In general, ionospheric scintillation
is described as a rapid or abrupt change in phase and amplitude of a radio
signal, while traversing through the ionosphere [14, 33, 34, 49, 61, 63]. This
phenomenon affects radio wave signals with frequencies ranging from 100 MHz to
4 GHz in general [11]. Several applications are affected by the ionospheric
scintillations including positioning, navigation, communication, and remote
sensing [2, 3, 4]. Owing to the variability of factors such as solar activity,
interplanetary magnetic field conditions, local electric field, conductivity,
convection processes, and wave interactions, ionospheric scintillation is
difficult to predict and model in the equatorial and low-latitude regions,
characterized by the Equatorial Ionization Anomaly (EIA) [13, 29, 28, 38].
Ionospheric scintillations are typically associated with high solar activity
and geomagnetic storms at high latitudes, while disruptions in and around the
doldrums are primarily caused by the EIA and ionospheric bubbles or Equatorial
Plasma Bubbles (EPBs), which cause extreme scintillation in radio wave
communications from satellites to receiver location on ground [21, 32, 67, 19,
68].

Several researchers have studied the phenomenon of ionospheric scintillation
and its causes and effects during extreme geomagnetic activity over the past
few decades using Global Positioning System (GPS) data. [61] was one of the
first studies to show, how the number of sunspots and geomagnetic latitude
affect radio star and satellite scintillations. The morphology of this global
phenomenon is clearly explained by [1]. It is possible to characterize and
comprehend the ionospheric disturbances that cause ionospheric scintillations
using the GPS data [21, 32, 39, 15]. During the magnetic storms of September
1999 and July 2000, studies by [10, 12] recorded depletion of ionization
density structures from middle and equatorial latitudes across the United
States, where plumes of greatly enhanced Total Electron Content (TEC) were
correlated with the erosion of the outer plasmasphere by turbulent sub-auroral
penetration electric fields. [8] was the first to report the effect of
$C/N_{0}$ fluctuations on GPS Dilution Of Precision (DoP) for minimum sunspot
number conditions under geomagnetically quiet conditions. [47] reported on the
occurrence of GPS cycle slips observed in a duration of exceptionally low
solar activity on October 8, 2009, correlated with the phenomenon of phase
scintillation, in a clear study from the Indian longitude sector. During
moderate to high solar activity levels in September 2011, a simple approach to
the causes of cycle slips of signals from satellite-based navigation systems
from two stations on and beyond the anomaly crest region was presented by
[54].

To understand the variabilities of the upper atmosphere and space weather
impacts on it over the geosensitive Indian longitude sector, the Indian Space
Research Organization (ISRO) conceived the regional navigation satellite
system: Navigation with Indian Constellation (NavIC). The space segment of
this system consists of seven satellites (a combination of Geostationary Earth
Orbit (GEO) and Geosynchronous Orbit (GSO) satellites). These satellites
transmit signals in the $L_{5}$ and $S_{1}$ bands, with carrier frequencies of
1176.45 MHz and 2492.028 MHz, respectively, in a 24 MHz bandwidth. It is
engineered to provide positional accuracy information not only to the Indian
users but also to the users within a 1500 km radius around its boundary,
defined by a rectangular grid spanning from 30oS to 50oN in latitude to 30oE
to 130oE in longitude [45]. The reliability of NavIC in exploring the upper
atmosphere has already been demonstrated by [40, 22, 5, 18, 46, 52, 53, 56,
17, 7, 58, 50, 16, 6]. The signal intensity and accuracy of NavIC satellites
have recently been confirmed by [57], to be reliable and used for ionospheric
scintillation studies. The study of observed ionospheric scintillation events
from GPS and NavIC measurements in the Bengaluru region was reported by [37].
Later, [51] reported two-station observations of ionospheric scintillations of
NavIC $L_{5}$ and $S_{1}$-band signals over the Indian subcontinent’s low-
latitude regions.

The present work, to the best of our knowledge, is the first long-term study
of low-latitude ionospheric scintillation, during low solar activity
conditions, using observations from both NavIC and GPS signals over a wide
area: Indore located near the northern crest of the EIA and Hyderabad located
in between the crest and the magnetic equator. This in-depth study would set a
path forward for a thorough understanding of the dynamic nature of the low-
latitude ionosphere during low solar activity conditions over the Indian
longitude sector. The approach for NavIC data for various other locations to
validate with multiple regions study was initiated during this work. But due
to the unavailability of the data which matches the analysis period and the
limited network of NavIC Rx, the analysis presented is limited to these
stations only.

##  2 NavIC and GPS Receiver - Data Analysis

In the Department of Astronomy, Astrophysics and Space Engineering (DAASE) of
the Indian Institute of Technology, Indore, a multi-constellation, multi-
frequency GNSS receiver (Septentrio PolaRxS Pro), capable to receive GPS
$L_{1}$ (1575.42 MHz), $L_{2}$ (1227.60 MHz), and $L_{5}$ (1176.45 MHz)
signals, as well as a NavIC receiver (ACCORD) provided by Space Applications
Centre, ISRO, intercepting GPS $L_{1}$, NavIC $L_{5}$, and NavIC $S_{1}$
(2492.028 MHz) signals, are operational since May 2016 and May 2017,
respectively. In addition to the Indore (Lat: 22.52oN, Lon: 75.92oE; Magnetic
dip: 32.23oN) station’s NavIC and GPS data, the present study also involves
the utilization of NavIC data from the station: Osmania University (OU),
Hyderabad (Lat: 17.40oN, Lon: 78.51oE; Magnetic dip: 21.69oN) as depicted in
Fig. 1 which shows the geographic locations of the GPS and the NavIC receivers
along with the Ionospheric pierce points (IPPs) as observed from each of the
locations.

The NavIC receiver generates raw log files that are post-processed to retrieve
the observables. The observables are only retrieved with the aid of the
Graphic User Interface (GUI) application (see for details in ACCORD-Original
Equipment Manufacturer (OEM) [45]). The GUI application receives, down-
converts and demodulates the transmitted satellite signals both at $L_{5}$ and
$S_{1}$ band frequencies. Most importantly, this GUI application generates
measurements precisely with respect to the external/internal trigger, such as
1 Pulse Per Second (PPS). In addition, the GUI application includes a GPS
receiver capable of receiving and processing $L_{1}$ Coarse/Acquisition (C/A)
code, centered at 1575.42 MHz signal, and generates measurements with respect
to the external/internal trigger. The receiver output provides the user
position by computing only $L_{5}$ , only $S_{1}$ , only $L_{1}$ , $L_{5}$ and
$S_{1}$ (Dual frequency) modes, and NavIC + GPS (Hybrid) mode depending on the
user command. The receiver uses Satellite-based Augmented System (SBAS)
corrections with the GPS automatically if SBAS correction parameters are
available. Control inputs to the system include a 10 MHz external reference
clock, 1-PPS signal; two RS232 interfaces for timing messages and National
Marine Electronics Association (NMEA) message format, and an Ethernet port for
monitoring, controlling, and commanding. The same Ethernet port is also used
to do the Software Upgrade (SWUG) of the receiver. The data update rate on the
NMEA port is 1 Hz and that of the Ethernet port is 5 Hz for GPS and 1Hz for
NavIC. The data obtained at this rate is logged in the format of comma-
separated variable (.csv) files for every three hours, using which an
automated pipeline has been developed which uses observables such as: Time
(TOW (s)), $C/N_{0}$, Azimuth, Elevation, Iono Delay, TEC and loss of lock at
the sampling frequency of 1 Hz to match the ACCORD generated NavIC observables
with Septentrio-generated GPS observables to generate the various plots of IPP
footprints, $S_{I}$, $S_{4_{C}}$, ROT and ROTI, as well as the $S_{coeff}$
values and the various distribution curves, plots, and values.

The Septentrio PolaRxS Pro is a state-of-the-art GNSS receiver that widens the
capabilities of studying the upper atmosphere by using a modern triple-
frequency receiver. The tracking engine is coupled with an ultra-low noise
OCXO (details given in POLARXS-OEM) frequency that generates and stores raw
high rate data which can be processed in real-time or in post-processing to
furnish 60s scintillation indices $S_{4}$ and $\sigma_{\phi}$, along with
other parameters like TEC, lock-time and the scintillation spectral parameters
such as the spectral slope and the spectral strength of the phase Power
Spectral Density (PSD) at 1 Hz for all visible satellites and frequencies. The
data from this receiver usually allows the user to utilize the data in the
following formats:

  * 1.

Highly compact and detailed Septentrio Binary Format (SBF) output.

  * 2.

NMEA v2.30 and v4.10 output format.

  * 3.

ISMR (Ionospheric Scintillation Monitoring) file generation using the provided
sbf2ismr utility.

In the present study, Data have been utilized during the whole period (25
months) of analysis, beginning from September 01, 2017, till September 30,
2019. Data are obtained in the .ismr format files generated every hour. A more
elaborate GPS data analysis procedure has been discussed in [20, 5].

Figure 1: The figure represents the locations of observation that are
presented in the work. The red stars in each of the maps represent the
receiver’s location. The eight shaped plots in each of the maps are the
ionospheric pierce point locations as observed from the GPS and NavIC receiver
locations for Indore (top) and NavIC receiver location for Hyderabad (bottom).

##  3 Observations of Scintillations

###  3.1 Criteria for scintillation

To extract scintillation events from the GPS data and based on evaluations of
non scintillation baseline indicators as illustrated by [31, 62] in support
with NOAA scales [65], 27 days are selected out of 607 days Fig: 2, 3 and 4)
analyzed spanning from September 01, 2017 to September 30, 2019 for the two
stations, the following criteria are given below:

  * 1.

The masking angle is set to $20^{o}$ to minimize multipath effects for the
elevation in data sets for both NavIC and GPS.

  * 2.

The data of $S_{4_{C}}$ is filtered in such a way that the threshold value is
0.3.

  * 3.

The event doesn’t qualify as a scintillation event unless it remains above
this threshold value for a period of at least 30 seconds, and any event which
initiates within an elapsed time of five minutes after the previous event is
not considered a separate event to avoid certain interference effects.

  * 4.

The $S_{4_{C}}$ events observed from multiple satellites simultaneously are
analyzed as separate events.

  * 5.

The events for which the loss-of-lock (LoL) has lasted for more than 120
seconds are evaluated for the analysis.

In this work, the LoL is defined as the ability of the receiver to lose track
of a transmitting satellite. During this time interval, the receiver will be
unable to log any phase or pseudorange observation for that duration.
Following the above set of criteria and analyzing the entire 25 months of
observations of NavIC Indore, only 27 days qualify as scintillation events
(Table 1). However, in the analysis, only 4% of the nights in the total
duration of the scintillations were detected and these events have been
selected based on the distribution along the studied period. The values of
sunspot number (SSN), the F10.7 solar radio flux (s.f.u.), and the Dst index
(nT), as well as peak values of $S_{4_{C}}$ and ROTI for the 27 days of
filtered data based on the selection criteria, are shown in Table 1.

Table 1: The $S_{4_{C}}$ occurrences observed for the 27 days, based on the
selection criteria of scintillation events, with the corresponding values of
the sunspot number (SSN), the F10.7 solar radio flux (s.f.u.), the Dst index
(nT), and the peak values of $S_{4_{C}}$ and ROTI over Indore and Hyderabad as
observed by NavIC and GPS over Indore (IITI) and Hyderabad (OU).

Date (yy-mm-dd) |  SSN |  F10.7 s.f.u |  Dst (nT) |  peak $S_{4_{C}}$ GPS IITI |  peak $S_{4_{C}}$ NavIC IITI |  peak $S_{4_{C}}$ NavIC OU |  peak ROTI GPS IITI |  peak ROTI NavIC IITI |  peak ROTI NavIC OU
---|---|---|---|---|---|---|---|---|---
17-09-08 |  88 |  188.5 |  -124 |  0.65 |  0.61 |  0.80 |  0.07 |  0.10 |  0.1
17-09-16 |  13 |  72.9 |  -30 |  0.30 |  0.35 |  0.29 |  0.10 |  0.81 |  0.06
17-09-27 |  37 |  91.3 |  -46 |  1.10 |  2.57 |  - |  0.11 |  0.55 |  -
17-10-12 |  0 |  69.9 |  -41 |  0.96 |  0.63 |  0.59 |  0.09 |  0.70 |  0.34
17-10-24 |  23 |  76.7 |  -28 |  0.33 |  1.20 |  0.33 |  0.09 |  0.32 |  0.30
17-10-25 |  23 |  77.9 |  -24 |  0.30 |  0.33 |  0.33 |  0.03 |  0.18 |  0.14
17-12-01 |  23 |  68.3 |  -8 |  0.32 |  0.34 |  0.32 |  0.08 |  0.07 |  0.064
17-12-04 |  23 |  66.4 |  -45 |  0.33 |  0.48 |  - |  0.1 |  0.77 |  -
17-12-14 |  0 |  69.8 |  -7 |  0.75 |  0.64 |  0.56 |  0.09 |  0.6 |  0.064
17-12-27 |  13 |  68.6 |  -11 |  0.33 |  1.39 |  0.31 |  0.09 |  0.19 |  0.06
18-02-16 |  12 |  69.8 |  -16 |  0.36 |  1.29 |  - |  0.13 |  0.91 |  -
18-02-19 |  0 |  67.5 |  -20 |  0.33 |  0.43 |  0.33 |  0.09 |  0.36 |  0.05
18-02-20 |  0 |  66.3 |  -28 |  0.35 |  0.98 |  0.30 |  0.1 |  0.25 |  0.06
18-02-21 |  0 |  68.7 |  -15 |  0.33 |  0.40 |  0.30 |  0.12 |  0.30 |  0.11
18-02-22 |  0 |  67 |  -13 |  0.30 |  0.31 |  0.31 |  0.10 |  0.13 |  0.05
18-02-24 |  0 |  66.8 |  -29 |  0.35 |  0.34 |  0.28 |  0.10 |  0.45 |  0.05
18-02-25 |  0 |  65.9 |  -15 |  0.35 |  0.45 |  0.36 |  0.10 |  0.15 |  0.06
18-02-27 |  15 |  66.6 |  -30 |  0.38 |  0.43 |  0.30 |  0.09 |  0.65 |  0.07
18-02-28 |  12 |  67.5 |  -20 |  0.40 |  0.38 |  0.37 |  0.11 |  0.09 |  0.12
18-03-01 |  12 |  66 |  -17 |  0.42 |  0.31 |  - |  0.11 |  0.65 |  -
18-03-02 |  11 |  66.6 |  -12 |  0.45 |  0.35 |  - |  0.1 |  0.12 |  -
18-03-13 |  0 |  67.7 |  -5 |  0.34 |  1.01 |  - |  0.12 |  0.24 |  -
18-03-22 |  0 |  68 |  -28 |  0.41 |  1.23 |  2.01 |  0.11 |  0.24 |  0.12
18-04-28 |  0 |  71.1 |  -1 |  0.34 |  0.36 |  - |  0.10 |  1.13 |  -
18-05-28 |  20 |  78.9 |  13 |  0.37 |  0.43 |  - |  0.1 |  0.12 |  -
18-06-05 |  0 |  73.4 |  -15 |  0.39 |  0.34 |  - |  0.11 |  0.18 |  -
18-09-26 |  0 |  69 |  -14 |  0.37 |  0.42 |  - |  0.10 |  0.55 |  -

Figure 2: The variation of F10.7 along with GPS SVID 1 observed values of peak
$S_{4_{C}}$ and ROTI by SVID 1 from Indore station for the available L band
frequencies, for the analysis period: September 2017 through April 2019.

Figure 3: The variation of F10.7 along with NavIC observed values of peak
$S_{4_{C}}$ and ROTI by PRNs 2, 5 and 6 from Indore station, for the analysis
period: September 2017 through April 2019.

Figure 4: The variation of F10.7 along with NavIC observed values of peak
$S_{4_{C}}$, and ROTI by PRNs 2, 5 and 6 from Hyderabad station, for the
analysis period: September 2017 through April 2018.

Fig. 2, 3 and 4 represent the overall variation of F10.7 flux along with peak
$S_{4_{C}}$ and ROTI for GPS Space Vehicle Identification (SVID) 1 over Indore
and the NavIC PRNs 2, 5 and 6 over Indore and Hyderabad respectively. The
blank spaces in these plots indicate data gaps during that period. It is found
that the intensity of scintillation, as indicated by $S_{4_{C}}$, gradually
increased during the equinoctial months (March–April, September–October). The
scintillation events were rarely observed in the other months (where the F10.7
flux values were low) of the period of analysis in both stations. In support
of the former observation, the corresponding values of ROTI, which are also
known as a proxy for the $S_{4_{C}}$ index, also show the same variation.
During the equinoctial months, the ROTI values for all the satellites of both
the constellations varied between 0.5 to 1 for most of the days. The seasonal
and solar activity dependence of the scintillation occurrence as observed
using NavIC over these regions is in agreement with that of previous studies
[59, 55, 69, 35].

###  3.2 Analysis of a typical scintillation event

This subsection describes a typical case of scintillation event observed on
September 08, 2017, based on the formulae provided to convert the $C/N_{0}$ to
scintillation index $S_{4_{C}}$ (equations 6 to 9 ) [63, 14, 67]. Furthermore,
the rate of change of slant TEC (ROT), introduced by [48], provides an
estimate of the ionospheric disturbances, by taking into account the average
standard deviation of the rate of slant TEC. This index is defined as the Rate
of TEC Index (ROTI). The formulae used to calculate values of ROT and ROTI are
presented in Appendix (equations 10 and 11 respectively). The $S_{4_{C}}$, the
ROT (dTEC/dt), the ROTI, and the scattering coefficients [27] (see 12) are
plotted as a function of LT (h) in Fig. 5, 6 and 7 for GPS and NavIC
observations over Indore and NavIC observations over Hyderabad respectively,
on September 08, 2017.

The dominant mechanism, in the onset of amplitude scintillations, is the Pre-
reversal Enhancement generated as a result of the enhancement in the vertical
E$\times$B drift due to the eastward electric field at the sunset terminator
(where the terminator implies a moving boundary between the day side and night
side regions on the Earth) generating EPBs and ESFs around this time [23, 64,
26]. [41] in their study have brought forward the relationship between the
onset of low-latitude scintillation and the sunset terminator. It is also to
be noted that the onset of scintillation at a given location represents the
time when the irregularities, that produce these scintillations, are initially
generated [36]. Fig. 5 (A to C) shows the onset times of amplitude
scintillation $S_{4_{C}}$ as observed from GPS SVID 1 at $L_{1}$, $L_{2}$, and
$L_{5}$ frequencies during the local time between 23:30 and 00:30 LT (h). Fig.
5 (D to F) and (G to I) show the ROT and ROTI values as observed from the same
SVID during the occurrence time of $S_{4_{C}}$. Fig. 5 (J to L) depicts the
variation of scattering coefficients ($S_{coeff}$) from all three frequencies
based on the deviations estimated from each of the frequencies of GPS. A clear
deviation from the mean value is observed, supporting the fact that signal
strength is affected during $S_{4_{C}}$ events, especially for the GPS
frequency combination ($S_{coeff}$) of GPS ${L_{2}L_{5}}$ and GPS
${L_{1}L_{5}}$. Similarly, Fig. 6 and 7 show the onset times as observed from
NavIC PRNs 2, 5, and 6 over Indore and Hyderabad, respectively. The onset
times were observed to be around 20:30 LT (h) and 19:30 LT (h) respectively
for Indore and Hyderabad, while the corresponding peak values of $S_{4_{C}}$
at these locations vary between 22:00 and 23:30 LT (h) for the PRNs 2, 5 and 6
of NavIC. It is clear from Fig. 5,6 and 7 that the $S_{4_{C}}$ oscillations
match with the oscillations observed in ROT, ROTI and $S_{coeff}$. Although in
Fig. 6 and 7 NavIC PRN 6 has no clear indicators of ROT and ROTI for both the
stations Indore and Hyderabad it is still clear that $S_{4_{C}}$ oscillations
match with the deviations of $S_{coeff}$. In addition to these observations,
it is to be noted that values of $S_{4_{C}}$ spiked up on these days,
particularly for the PRNs 2, 5, and 6. This trend is noted during the whole
period of the analysis presented in this work for the two stations.
Additionally, the oscillating values observed in the $S_{S_{1}L_{5}}$ match
with the fluctuations of $S_{4_{C}}$. The work presented here clearly shows
that NavIC amplitude fluctuations present a pattern in the onset time of
scintillation detection. During the period of analysis, for all the days, the
difference in the onset time of scintillation was mostly during the post-
sunset period when compared to pre-sunrise events. However, as shown by [59],
a similar pattern in the scintillation onset time has been observed for the
GPS signals over the Brazilian sector, which is near to southern crest of EIA,
with an average onset period of 30 to 60 minutes before the onset time
observed in this work. The peak values and onset time of scintillations at
various stations covering the Indian region using GPS and Digi-sonde data,
reported by [55, 60] showed that the scintillation onset time occurred between
19:30 LT and 23:30 LT, closely matching the current results. In support of the
present observations, the variation of the onset time of scintillation as
presented in [69, 35] showed that measurements of scintillation using GPS
satellites cover a wide longitude region, and the onset time difference of
scintillation during a specified season is linked with the different locations
where scintillation was initially generated, especially near the northern
crest of EIA.

Figure 5: The $S_{4_{C}}$, ROT, ROTI, and $S_{coeff}$ variations as observed
from Indore station during during 21:30 LT (h) of September 08, 2017 to 00:30
LT (h) of September 09, 2017 for GPS SVID 1 for all three L-band frequencies.

Figure 6: The $S_{4_{C}}$, ROT, ROTI and $S_{S_{1}}{L_{5}}$ variation on
September 08, 2017 as observed from Indore station during 17:30 LT (h) of
September 08, 2017 to 05:30 LT (h) of September 09, 2017 for NavIC PRNs 2,5
and 6.

Figure 7: The $S_{4_{C}}$, ROT, ROTI and $S_{S_{1}}{L_{5}}$ variation on
September 08, 2017 as observed from Hyderabad station during 17:30 LT (h) of
September 08, 2017 to 05:30 LT (h) of September 09, 2017 for NavIC PRNs 2,5
and 6.

##  4 Characterizing Amplitude Scintillation

After identifying the scintillation events over this period of 25 months, we
proceeded towards characterizing the scintillation events observed from NavIC
Indore and validated by NavIC Hyderabad and GPS Indore. To characterize
amplitude scintillations at low-latitudes, earlier studies [24, 25] found
evidence to evaluate statistical models. The Nakagami-m distribution was found
to be the optimum fit for the data set in the study results. A decade later
according to [9], theoretical Nakagami-m distribution was found to be the best
for explaining different levels of fade. In the early 1990s, this was one of
the first studies in the Indian region, with a C and L band transionospheric
scintillation. [30] proposed a simple model for simulating equatorial
transionospheric radio wave scintillation, and this study concluded that the
Rice distribution was considered to be the best fit for the experimental data.
Following these experiments, [44, 43, 42] identified the benefits of using a
two-parameter model, recommending the use of the $\alpha-\mu$ and Nakagami-m
distributions. The studies proposed a model based on the $\alpha-\mu$
distribution parameters and taking into account the amplitude and phase
scintillation correlation [66]. The numerous findings presented in this study
clearly illustrate how, under the influence of ionospheric conditions, a
receiver experiences different error values leading to a specified
scintillation level. $S_{4_{C}}$. [66] introduced a $\alpha-\mu$ distribution
which in general is a fading model, which explores the non-linearity of the
propagation medium and is a revised form of the Stacy (generalized Gamma)
distribution. [44] proposed the use of this distribution for modeling
ionospheric amplitude scintillation.

The model is described by the coefficients $\alpha$ and $\mu$ of the
normalized amplitude envelope of the received signal ($r$), where $\alpha$ is
the modulus of the sum of the multipath components and $\mu$ is the number of
multipath components and the $\alpha-\mu$ probability density function,
assuming that the average signal power (or intensity) $r^{2}$ is equal to 1
for $E[R^{2}]=1$ is defined as:

$f(r)=\frac{\alpha{r^{\alpha\mu-1}}}{{\zeta^{\alpha\mu/2}}{\Gamma(\mu)}}exp\big{(}-\frac{r^{\alpha}}{\zeta^{\alpha/2}}),$
(1)

where in

$\zeta=\frac{\Gamma(\mu)}{\Gamma(\mu+2/\alpha)}$ (2)

$\Gamma(.)$ is the gamma function. [66] used this model to describe the mobile
communication channel, assuming that the received signal is the consequence of
a cluster of multipath waves propagating in an heterogeneous environment. The
most interesting feature of this model is that it can simulate a number of
different scenarios depending on the values of its variables, for instance,
the $\alpha-\mu$ distribution can be transformed into a Nakagami-m
distribution ($\alpha$ = 2 and $\mu$ = m, where m is the single parameter of
the distribution), a Rayleigh distribution ($\alpha$ = 2 and $\mu$ = 1), or a
Weibull distribution (with two parameters $\alpha$ and $\gamma$
=$\zeta^{1/2}$, for $\mu$ = 1).

As described in Appendix, $S_{4}$ index characterizes the strength of
amplitude scintillation where the intensity of received signal $I=|r^{2}|$.
The Nakagami-m parameter in the Nakagami distribution can be related to
$S_{4}$ index by the following notation where $m={1}/{S_{4}^{2}}$ and the
relation to estimate this is given below

$m=\frac{{E}^{2}({r}^{2})}{{{E}({r}^{4})}-{E}^{2}({r}^{2})}$ (3)

The $\alpha$ and $\mu$ coefficients can be calculated based on an experimental
study conducted by [66].

$\frac{{E}^{2}({r}^{\beta})}{{{E}({r}^{2\beta})}-{E}^{2}({r}^{\beta})}=\frac{\Gamma^{2}(\mu+\beta/\alpha)}{\Gamma(\mu)\Gamma(\mu+2\beta/\alpha)-\Gamma^{2}(\mu+\beta/\alpha)}$
(4)

The left-hand side of the equation (4) can be derived from field data for
randomly chosen values of the parameter $\beta$, which provides the order of
the system of $r$ to be determined. In fact, when the left-hand side of
equation (4) for $\beta$ = 2 is compared to the right-hand side of equation
(7), the following results are obtained:

${S_{4}}^{2}=\frac{\Gamma(\mu)\Gamma(\mu+4/\alpha)-\Gamma^{2}(\mu+2/\alpha)}{\Gamma^{2}(\mu+2/\alpha)}$
(5)

If $\alpha$ = 2 in the above equation (5) and the properties of the gamma
function are used, the value of the left-hand side of the same equation
implies $1/\mu=1/m$, which is the condition for the Nakagami-m distribution
([44]).

Applying the same to characterize amplitude scintillation, analysis for a
chosen event of scintillation on September 08, 2017, where the $C/N_{0}$ value
has significantly dropped below 30 dB-Hz and has approached zero between 22:00
to 0:00 LT (h) as observed from both the constellations, is considered. During
the event, only SVID 1 from the GPS constellation was able to capture the
intense value of amplitude scintillation. The corresponding signal intensity
fading (20log($r$)) for Indore GPS SVID 1 during the $C/N_{0}$ drop period,
lasted for the period of 3400, 600, and 600 seconds for $L_{1}$, $L_{2}$, and
$L_{5}$, respectively. The $S_{4_{C}}$ index as a function of local time is
shown for each of the frequencies in Fig. 8 (a, d and g) respectively.

The values where the amplitude scintillation peaks are marked (in a
rectangular box) to indicate the fading sample that is used to calculate the
signal intensity fading (20log($r$)) as a function of time in seconds in Fig.
8 (b, e, and h). The corresponding values of $\alpha-\mu$ and Nakagami
distribution for the scintillation occurrences as a function of 20 log ($r$)
are shown in Fig. 8 (c, f and i) respectively for $L_{1}$, $L_{2}$, and
$L_{5}$ frequencies. The scintillation occurrences are the empirical data in
the plots for which the distribution curves have been plotted in Figs. 8,9 and
10. In the case of GPS-SVID 1, though the rectangular box appears to be for a
longer time interval, the instances of discontinuous data at $L_{2}$ and
$L_{5}$ frequencies reduce the fading sample to be less than $L_{1}$. The
NavIC (Indore and Hyderabad) observations on the same day for the $S_{4_{C}}$
indices for PRN 2, 5 and 6 are shown in Fig. 9 and 10 (a, d and g) along with
the sample of data points chosen to calculate corresponding signal fading for
these PRNs in Fig. 9, 10 (b, e and i). The distribution curves for
$\alpha-\mu$ and Nakagami for the occurrences (empirical data) here in the
case of NavIC are plotted in Fig. 9, 10 (c, f and i) respectively. The sample
windows in the above observations and corresponding Fig. 8, 9 and 10 show how
the severity of fading occurrences increases as the value of $S_{4_{C}}$
rises. The results for the values of $\alpha-\mu$ parameters also show that
$\alpha$ increases with the increase in $S_{4_{C}}$. This depicts how
challenging it is for the receiver to retain the lock as the signal quality
degrades. As a result, defining the statistics of these fades is important for
estimating the effects of these phenomena and supporting the development of
signal processing techniques and positioning algorithms that will reduce these
effects in the receiver. The observations presented will be used as the
starting points for further studies related to our evaluation of positional
accuracy with the aid of NavIC under disturbed as well as quiet-time
ionospheric conditions.

Figure 8: Scintillation event on September 08, 2017, as observed by GPS SVID 1
$L_{1}$, $L_{2}$, and $L_{5}$ signals over Indore. Panels a, d, and g show
variations of $S_{4_{C}}$ for a portion of the day. The black rectangular box
denotes the epochs when scintillation was detected. Panels b, e, and h shows
fading signal intensity for the period marked with a rectangular box in panels
a, d and g. Panels c, f and i show the calculated occurrences along with the
$\alpha-\mu$ distributions and the Nakagami-m distribution (refer to equations
1 to 5) Figure 9: Scintillation event on September 08, 2017, as observed by
NavIC PRNs 2,5 and 6 signals over Indore. Panels a, d, and g show variations
of $S_{4_{C}}$ for a portion of the day. The black rectangular box denotes the
epochs when scintillation was detected. Panels b, e, and h shows fading signal
intensity for the period marked with a rectangular box in panels a, d and g.
Panels c, f and i show the calculated occurrences along with the $\alpha-\mu$
distributions and the Nakagami-m distribution (refer to equations 1 to 5)
Figure 10: Scintillation event on September 08, 2017, as observed by NavIC
PRNs 2, 5, and 6 signals over Hyderabad. Panels a, d, and g show variations of
$S_{4_{C}}$ for a portion of the day. The black rectangular box denotes the
epochs when scintillation was detected. Panels b, e, and h shows fading signal
intensity for the period marked with a rectangular box in panels a, d and g.
Panels c, f and i show the calculated occurrences along with the $\alpha-\mu$
distributions and the Nakagami-m distribution (refer to equations 1 to 5)

##  5 Discussions and Conclusion

A systematic study of the variable low-latitude region around the EIA and the
magnetic equator is always essential to probe the ionosphere as well as its
impact on other aspects of communication and navigation. Here, a study in low
solar activity conditions using NavIC is presented. It should be noted that in
a previous study, see [5], it has been shown that NavIC has significant
advantages over the GPS for ionospheric studies in the Indian subcontinent
region. These observations, when combined with the GPS constellation, and the
additional ray path through the ionosphere provided by this system, help to
better observe space weather effects. The observations presented show NavIC’s
efficiency, in studying space weather, which has been evaluated at Indore
using 27 critical days out of 607 days of observation. Detailed case studies,
from two different stations (Indore: located near the northern crest of EIA,
and Hyderabad: located near the magnetic equator), have been performed to
validate the reliability of NavIC to study the ionosphere during space weather
events. It has been found that the variation of $S_{4_{C}}$ registered from
the GPS closely matches that of NavIC $S_{4_{C}}$. It is even more interesting
to observe the post-sunset ionospheric response of NavIC PRNs 2, 5, and 6
w.r.t the $S_{4_{C}}$ index. The values of NavIC $S_{4_{C}}$ and the ROT, as
well as ROTI indices, have a clear signature of the occurrence of ionospheric
scintillation. The scintillation onset times have been observed to be
occurring around 19:30 LT (h) and 20:30 LT (h) for Hyderabad and Indore,
respectively, and the peak values of $S_{4_{C}}$ to be occurring between 22:00
and 23:00 LT (h). The $S_{4_{C}}$ proxy parameters: ROT, peaked at 0.2 and
ROTI, peaked at 0.1. The fluctuations in Fig. 3 are more when compared to the
ones in Fig. 4 and the reason behind the variation in fluctuations is still
not known and can be answered well only when such an analysis is carried at
various locations with NavIC and GPS. The results obtained in our study are in
complete agreement with similar works presented in previous literature by [11,
38, 13, 32, 51, 10]. One of the important outcomes of our study is that it
shows, for the first time, the existence of a strong correlation between NavIC
signals from $L_{5}$ and $S_{1}$ (see Fig. 5, 6, and 7). We have shown the
same using the scattering coefficients analysis where it is found that the
scattering is minimum between the $L_{5}$ and $S_{1}$ signals of NavIC. This
is in contrast with the previous study done by [27] where they used $L_{1}$
and $L_{5}$ signals of GPS. Furthermore, studies presented here using
empirical scintillation data show that the amplitude distribution is well
modeled as Nakagami-m and $\alpha-\mu$ distributions as presented in [25, 9,
30, 44, 43, 42]. These models have been validated in comparison tests with
empirical scintillation data for various PRNs of NavIC. A follow-up work, from
this study based on a detailed characterization of the scintillation events
during these 25 months using the $C/N_{0}$ observations from both NavIC and
GPS, has also been performed. While the work presented here is limited to two
NavIC stations, similar comparative studies with NavIC at different stations,
under variable ionospheric conditions, will be carried out in the future to
validate our results over large geographic locations of the Indian
subcontinent.

## Acknowledgments

DA acknowledges the INSPIRE fellowship from the Department of Science and
Technology, which she used to pursue her research. SC acknowledges ISRO’s
Space Applications Center (SAC) for offering an NGP-17 research fellowship
during his Ph.D. and the Department of Space for providing a fellowship during
his post-doctoral research. The authors also thank SAC, ISRO for providing the
NavIC ACCORD receiver to the Department of Astronomy, Astrophysics, and Space
Engineering, IIT Indore, under the NGP-17 project. The authors wholeheartedly
thank Dr. P. Naveen Kumar for providing NavIC Osmania University (OU)
Hyderabad station data. DA is extremely thankful to Harsha Avinash Tanti and
Anshuman Tripathi for the fruitful discussions and moral support throughout
the analysis period.

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##  Appendix A Calculations of $S_{4_{C}}$ index, the ROT, the ROTI, and the
Scattering Coefficients

The study presented in this paper utilizes Carrier to Noise density
($C/N_{0}$) (dB-Hz) measurements of NavIC satellite signals to estimate signal
intensity ($S_{I}$) as its primary data for the $S_{4_{C}}$ index values.

$S_{I}=10^{0.1*C/N_{0}}$ (6)

where the value of $S_{I}$ is denoted as the intensity of the signal. The
$S_{4}$ index is then calculated defined as the signal’s normalised variance
of intensity, which is expressed as:

${S_{4}}=\sqrt{\frac{\langle{S_{I}}^{2}\rangle-{\langle
S_{I}\rangle}^{2}}{{\langle S_{I}\rangle}^{2}}}$ (7)

where $S_{I}$ is the intensity of the signal. Thus calculated $S4$ index still
requires a correction factor as the $C/N_{0}$ value of any radio signal is
inclusive of ambient noise and in order to calculate the $S_{4}$ by applying
correction as below gives us ambient noise free $S_{4}$ index and is denoted
as $S_{AN}$ index

$S_{AN}=\sqrt{\frac{100}{C/N_{0}}}\Big{[}1+\frac{500}{19C/N_{0}}\Big{]}$ (8)

Now subtracting equation (8) from equation (7) and the real estimate for
$S_{4_{C}}$ is obtained as shown below

$S_{4_{C}}=S_{4}-S_{AN}$ (9)

The ROT and ROTI values [Pi et al., 1997] are calculated as given in equations
below where in equation (10) $STEC_{r+1}$ and $STEC_{r}$ are slant TEC at r+1
and r time epochs; $\Delta t_{r}$ time interval; usually the unit of ROT is
TECU/min and in equation (11) where $\langle ROT\rangle$ denotes averaging ROT
during N epochs.

${ROT}=\frac{{STEC_{r+1}}-{STEC_{r}}}{\Delta t_{r}}.$ (10)

${ROTI}=\sqrt{\langle{ROT}^{2}\rangle-{\langle ROT\rangle}^{2}}$ (11)

A scattering coefficient [Goswami et al., 2017] across a pair of frequencies
was later defined as the difference of $C/N_{0}$ fluctuations normalized with
respect to the amount of those fluctuations, based on these computed $C/N_{0}$
deviations. The formula for scattering coefficient is given by 12 where a is
the $C/N_{0}$ deviations calculated from one frequency is substituted and b is
the $C/N_{0}$ deviations calculated from second frequency respectively. Here
$S_{a,b}$ is a dimensionless quantity with values near zero, suggesting a good
similarity between the signals. Here, the $C/N_{0}$ deviations were determined
by subtracting the moving averaged values over a ninety-minute running time
period from the instantaneous $C/N_{0}$ calculation.

$S_{a,b}=\Big{[}\frac{a-b}{a+b}\Big{]}$ (12)

The $C/N_{0}$ deviations of NavIC $S_{1}$ and $L_{5}$ signals are calculated.
The scattering coefficients as observed from NavIC signals $L_{5}$ and $S_{1}$
show a very strong relationship during the quiet period of the ionosphere and
even during the scintillation period the scattering coefficients values have
not oscillated beyond the value of 0.1 for both observations over Indore and
Hyderabad Fig. 6 and Fig. 7 of Section 4.