University of Southern California Dissertations and Theses

Empirical approach for estimating the ExB velocity from VTEC map

(USC Thesis Other)

Empirical approach for estimating the ExB velocity from VTEC map

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content

Empirical approach for estimating the ExB velocity from VTEC map
By
Xi Ao

A Thesis Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
(APPLIED MATHEMATICS)
June 2014

ii
Acknowledgements

I would like to express my gratitude to my thesis advisor Professor Dr. Chunming Wang for the
continuous support of my work and for his motivation, enthusiasm, and immense knowledge.
Thank you for always being abundantly helpful, understanding and encouraging that complete
my understanding of ExB velocity. Dr. Chunming Wang not only guided me throughout this
thesis, but also gave me many lessons, which will be invaluable in my future study.
This work would not have been possible without the support of all the previous
researchers. Their vast knowledge in the domain of the ionospheric research was an endless
source of learning for me.
One last thank you goes to my family and friends who always supported me during these
years, and especially to my parents who have continuously inspired me, ignited my love for
mathematics and implanted in me the believe that science is the key factor to solve all problems.

iii
Table of Contents
Acknowledgements ......................................................................................................................... ii
Table of Contents ........................................................................................................................... iii
List of figures ................................................................................................................................. iv
Abstract ........................................................................................................................................... v
Chapter 1: Introduction ................................................................................................................... 1
1.1 The Earth’s ionosphere and its importance ......................................................................... 1
1.2 The mathematical model of the ionosphere and the key drivers ......................................... 3
Chapter 2: Empirical approach for estimating the ExB velocity from VTEC map ........................ 9
2.1 Retrieval of the equatorial anomaly gap (EAP) from VTEC map ...................................... 9
2.2 Sensitivity and Validation study ....................................................................................... 22
2.3 Correlation of ExB velocity and EAP ............................................................................... 24
Chapter 3: Conclusion .................................................................................................................. 28
References ..................................................................................................................................... 29

iv
List of figures
Figure 1: illustration of the definition of geomagnetic equator .................................................... 12
Figure 2: location of geomagnetic equator as of April 2011 ........................................................ 13
Figure 3: illustration of the definition of meidional line ............................................................... 14
Figure 4: location of the meridional line at longitude = -120 as of April 2011 ............................ 15
Figure 5: illustration of one step of the iterative method .............................................................. 17
Figure 6: VTEC map along the meridional line at longitude = -120 as of 2011/04/26 00:00 UT 18
Figure 7: definition of EAP in meridional line at longitude = -120 as of 2011/04/26 00:00 UT . 19
Figure 8: some positions of estimation method in meridional line at longitude = -120 as of
2011/04/26 00:00 UT .................................................................................................................... 20
Figure 9: maximum of EAP by estimation vs AP as of 2011/04/26 00:00 UT ........................... 22
Figure 10: maximum of EAP by estimation vs F10.7 as of 2011/04/26 00:00 UT ...................... 23
Figure 11: maximum of ExB velocity vs AP as of 2011/04/26 00:00 UT ................................... 24
Figure 12: maximum of ExB velocity vs F10.7 as of 2011/04/26 00:00 UT ............................... 25
Figure 13: maximum of Estimation EAP vs maximum of ExB velocity by fixing AP as of
2011/04/26 00:00 UT .................................................................................................................... 26
Figure 14: maximum of Estimation EAP vs maximum of ExB velocity by fixing F10.7 as of
2011/04/26 00:00 UT .................................................................................................................... 27

v
Abstract
For the development of wireless communication, the Earth’s ionosphere is very critical. A
Matlab program is designed to improve the techniques for monitoring and forecasting the
conditions of the Earth’s ionosphere. The work in this thesis aims to modeling of the dependency
between the equatorial anomaly gap (EAP) in the Earth’s ionosphere and the crucial driver, ExB
velocity, of the Earth’s ionosphere.
In this thesis, we review the mathematics of the model in the eleventh generation of the
International Geomagnetic Reference Field (IGRF) and an enhancement version of Global
Assimilative Ionospheric Model (GAIM), GAIM++ Model. We then use the IGRF Model and a
Vertical Total Electron Content (VTEC) map from GAIM++ Model to determine the EAP in the
Earth’s ionosphere. Then, by changing the main parameters, the 10.7cm solar radio flux (F10.7)
and the planetary geomagnetic activity index (AP), we compare the different value of the EAP in
the Earth’s ionosphere and the ExB velocity of the Earth’s ionosphere. At last, we demonstrate
that the program can be effective in determining the dependency between the EAP in the Earth’s
ionosphere and the ExB velocity of the Earth’s ionosphere.

1
Chapter 1: Introduction
1.1 The Earth’s ionosphere and its importance
Gases within a layer of the atmosphere at altitude between 50km to 2000km consist of the earth’s
ionosphere. The characteristics of this layer of atmosphere are the presence of free electrons and
ions that transmit, refract, and reflect radio waves, which allow those waves to be thoroughly
transmitted over great distances around the earth. The existence and importance of ionosphere
have been realized along with the development of wireless communication, and especially in the
area of radio broadcast. Ionized gases can provide a transmission channel for radio waves over a
long distance throughout certain wavelength. In the modern society, social, economical, and
national defense activities have been more and more highly influenced by the impact of the
earth’s ionospheric conditions. Additionally, as the development of international trade, the
quality of telecommunication has been extremely vital in business negotiations all over the world.
As a result, the monitoring of ionospheric conditions is critical for many technical applications:
the accuracy of global position system (GPS)-based navigation systems, high frequency (HF)
radio wave, and its propagation.
Densities, temperature and velocity of neutral gases, ion species and free electrons are
the basic quantities for ionospheric dynamics. The dynamic in ionosphere has a great impact on
the interaction between the ions and free electrons with both geomagnetic and electric fields.
And the solar radiation, ionizing the neutral gases, is the most significant external driving force
for ionosphere. And in general, we have the following simplifying approximations: (1) The
impact of ion species on the neutral gases is weak due to the relatively low number of ionized
particles, which is compared to the density of neutral gases. Consequently, neutral gases are
considered as independent of ionized gases in most researches of ionosphere models. In other

2
words, the impact of reverse interaction is eliminated; (2) Under the influence of motions of free
electrons and ions, the changes of electric and magnetic fields are considerably small, especially
when comparing with the changes of the magnetic field caused by solar activities. As a
consequence, it is more effective to regard both magnetic and electric fields as driving forces; (3)
As a direct function of solar angle and geomagnetic activity, the thermal balance of ionized gases
indicate that the temperatures of ionized gases can be assumed to be known.
The earth’s ionosphere is in a constant state of flux. So its "mirror property" can be
sometimes limited, though it seems to be a natural ‘mirror’ for radio waves. Just like terrestrial
weather, ionospheric properties constantly change. This ionospheric variability, which is called
space weather, can cause unreliability in both ground- and space-based communications that
usually depend on ionospheric reflection or transmission. Space weather variability have an
impact on how the earth’s ionosphere attenuates, reflects, absorbs, refracts, and changes the
phase, the propagation, and some other amplitude characteristics of radio waves. These weather-
based changes may caused by certain space weather conditions such as: (1) variability of solar
radiation which enters the upper atmosphere; (2) the solar plasma which enters the earth's
magnetic field; (3) the gravitational atmospheric tides which are produced by the sun and moon;
and (4) the vertical swelling of the atmosphere resulting from daytime heating of the sun. Space
weather is also significantly influenced by solar flare activity, the tilt of the geomagnetic field of
the earth, and abrupt ionospheric changes due to events such as geomagnetic storms.
Therefore, the inherent reflectivity of the earth’s ionosphere is a natural gift that humans
can use to create long-range communications to connect distant points on the globe. However,
natural variability in the ionosphere will reduce the reliability of our communication systems,
which depend on ionospheric refraction and reflection, primarily on HF. For the most part, some

3
higher frequency communications such as extremely high frequency (EHF), super high
frequency (SHF), and ultra high frequency (UHF) bands are transmitted through the earth’s
ionosphere without distortion. These bands are however limited to degradation resulting from
ionospheric scintillation, a phenomenon induced by abrupt changes in electron density along the
signal path, causing signal fade due to rapid signal path changes and defocusing of the amplitude
and/or phase of the signal.
Since the interaction between radio waves and free electrons in earth’s ionosphere will
lead to reflection of radio waves and delay in the wave propagation, monitering and predicting
ionospheric variability and understanding its influence on the transmission and reflection of
electromagnetic radiation must be a much-studied field of scientific inquiry. In addition, the
fundamental knowledge of ionospheric physics is very critical to complete the modeling of the
Earth’s ionosphere, because the more we explored the essence of ionospheric physics, the better
we can understand the assimilation results. Improving our ability to observe, model, and forecast
ionoshperic conditions will substantially improve both ground- and space-based communication
systems.

1.2 The mathematical model of the ionosphere and the key drivers
Because of the importance of knowing the changes of the Earth’s ionosphere, monitoring and
forecasting of its conditions is quite significant. To improve these techniques, the most important
part lies in modeling the changes of the ion and electron density in the Earth’s ionosphere and
assimilating remote sensing measurements into a numerical model. Equations of fluid for

4
chemically active plasma in the earth’s ionosphere, which consist of the governing equations of
the ionosphere, form a system of coupled nonlinear convection-diffusion equations.
According to past literatures, a large amount of first-principles ionospheric models have
been studied before. Parameters such as solar Extreme Ultraviolet (EUV) radiation,
thermospheric densities, electric fields, composition and temperature, and auroral precipitation
have great influence on the success of these models. Inputting these drivers into the collisional
plasma hydrodynamic equations, the ionization, dynamical and chemical processes can be
controlled. However, this generates a challenge in modeling ionospheric conditions because of
the difficulty in obtaining the required information of the model inputs. To solve this problem,
the researchers have tried to use measured ionospheric TEC, at some middle latitudes for
improvements. For example, JPL team built an ionospheric model named by Global Ionospheric
Map (GIM) for researching in the ionosphere. GIM is generated hourly using slant TEC data,
which is derived from the measurements of signal delays between the ground receivers and the
GPS satellites. JPL team has extensively calibrated and validated GIM against other independent
measurements of the ionosphere of the earth.
Actually, the original theoretical model was expected to have a better result after
estimating perturbed ExB drift and meridional wind, as asserted by Pi et al. in 1993. However,
these improvements were designed based on investigations of the physics processes rather than
assimilating data. Data driven techniques, including computerized ionospheric tomography (CIT)
and TEC mapping respectively developed by Mannucci in 1998 and Iijimal et al. in 1999, are
proven to be an alternative to the first-principles modeling. Throughout the line of sight TEC
measurements, these data-driven techniques offer mathematical methods to estimate the
ionospheric state variables.

5
The process of reconciling a first principle-based model of a complicated system with
some actual observations of the system to obtain accurate estimation of the initial state is called
data assimilation. It is also widely used in the area of measuring atmospheric temperature,
moisture, pressure and velocity. University of Southern California (USC) and a collaborative
research team of scientists from the Jet Propulsion Laboratory (JPL) started and kept using data
assimilation techniques for monitoring and forecasting ionospheric conditions since 1999.
Supported by the 1999 Multidisciplinary University Research Initiative (MURI) program, the
USC/JPL team created the first version of Global Assimilative Ionospheric Model (GAIM).
GAIM is designed to demonstrate consistent information on ionosphere, which can be
derived by using vastly different measurements. These measurements of several types: ground-
based total electron content (TEC), which is along the line of sight between the GPS satellites
and ground-based receivers. Another type is TEC along the line of sight between a Low Earth
Orbit (LEO) satellite and GPS satellites. The third type is Vertical Total Electron Content
(VTEC), which majorly come from space-based altimeters. The fourth category is the in-situ
electron density, which is measured by defense meteorological satellite program (DMSP)
satellite. The fifth category is ionosonde data, which provides a vertical electron density profiles
from ground-based on radar.
It is quite clear that the main part of these measurements depends on space-based
instrument. More importantly, the rapid increase in the available measurements of the ionosphere
is an effective stimulator for ionospheric data assimilation. In 2006, a family of 6 LEO satellites
(COSMIC) were launched and the number of GPS occultation were measured in order to
complete global cover of the ionosphere.

6
Notably, important ionospheric physics is lacking in the early version of the model. Even
though the original model was sufficient as a prototype for GAIM, the developing a complete
multi-ion version of the basic ionospheric model can be extended. However, the biggest obstacle
lies in multi-ion version of the ionosphere model equations, which is a nonlinear degenerated
system of parabolic partial differential equation. The single ion version of the model equations is
however a scale linear partial differential equation. The differences of the measurements
methods between the above two models cause the significant problem for developing a new
complete multi-ion version of the basic ionospheric model. Additionally, the maintenance of the
stability and positiveness of the solution is essential for approximating the solution of the
nonlinear system of partial differential equation.
In 2008, Vardan Akopian developed a new version of GAIM, called GAIM++. As an
enhancement of the earlier version of GAIM, GAIM++ Model enables us to examine the
interactive processes, which occur in the plasma of the Earth’s ionosphere throughout all most
relevant ion species. Additionally, this approach provides a solution of the ionospheric model
equations, which leads to physically meaningful and stable ion densities. Compared with other
existing numerical models of the ionosphere, GAIM++ has the usual features, which are
particularly useful for data assimilation. Allowing efficient implementation of the four-
dimensional variational approach (4DVAR) approach and Kalman filter approach for data
assimilation, the parameterization of the driving forces and the solution to the adjoint equation
highlights the function of the new version.
By adopting a fixed three-dimensional Eulerian grid, which follows a geomagnetic dipole
configuration, GAIM++ has been developed to improve the modeling of ionospheric condition.
In order to minimize the difference between modeled and measured line-of-sight TEC, A

7
4DVAR with the adjoint technique is presented via nonlinear least square minimization. As a
basis for the development of a 4DVAR data assimilation approach to help estimate the driving
forces in the earth’s ionosphere, GAIM++ uses the adjoint approach for computing the gradient
of performance functional. At the same time, the numerical method can accurately reproduce all
major characteristics of the ionosphere and the 4DVAR approach can be extremely effective in
determining the crucial drivers of the ionosphere of the earth. In addition, experience with
4DVAR is very critical to tune and calibrate GAIM++ to produce reliable forecast.
In GAIM++, the estimation technique is used for adjusting three physics drivers: the solar
production, the ExB velocity and the neutral wind.
The neutral wind is a driving force affecting equatorial anomaly. Notably, the neutral
wind follows a magnetic field line. The presence of convection will skew the symmetric process
of diffusion in one-dimensional convection-diffusion equation. For example, a strong southward
neutral wind gives preference to southward diffusion relative to northward diffusion of ions
along a flux tube, which leads to higher density and VTEC south of the equator relative to the
high-density region north of the equator.
Considering solar production, it is notably that the production rate in ionosphere is the
least predictable element. Because of the impact of the combination of solar radiation and neutral
density, ion production rate is not in general considered as independent. Since neutral density
determines the optical depth that reduces the intensity of solar radiation and chemical reaction
rates, in the modeling approach, instead of separating the contributions from solar radiation and
neutral density changes, direct estimation of ion production is used to allows the researchers to
directly change the ion density by changing production rate.

8
ExB velocity is the driver we investigate in this thesis. In fact, the distance between two
high-density regions of equatorial anomaly is the most widely used characteristic of the earth’s
ionosphere for estimating driving forces, and the strength of ExB velocity is correlated to this
distance. If ExB velocity increases, Ions will be pushed to increasingly higher flux tubes, which
will lead to a larger distance between two high VTEC regions around the geomagnetic equator.
X. Pi et al. in 2003 conducted an observation system simulation experiment (OSSE) for an
estimation of ExB velocity by using GAIM. In this thesis, we introduce a method for estimating
the ExB velocity: an empirical approach for estimating the ExB velocity from VTEC map.

9
Chapter 2:
Empirical approach for estimating the ExB velocity
from VTEC map
2.1 Retrieval of the equatorial anomaly gap (EAP) from VTEC map
To define Equatorial Anomaly Gap, we should introduce the eleventh generation of the
International Geomagnetic Reference Field (IGRF), which was adopted by the International
Association of Geomagnetism and Aeronomy Working Group V-MOD. It improves the previous
IGRF model with a main field model for epoch 2010, a linear predictive secular variation model
from 2010 to 2015, and a definitive main field model for 2005.
On and above the Earth’s surface, the IGRF model determines the geomagnetic field B(r, θ, φ, t),
which is produced by internal sources by using a scalar potential V(r, θ, φ, t). We then have B =
−∇V where V is a finite series having several numerical Gauss coefficients 𝑔
!
!
, ℎ
!
!

(conventionally given in units of nanotesla, denoted by nT):
V(r, θ, φ, t) = a
∑
n=1
N

∑
m=0
n

⎝
⎜
⎛
⎠
⎟
⎞
a
r
n+1
[g
m
n
(t)cos(mϕ)+h
m
n
(t)sin(mϕ)] × P
m
n
cos(θ).
(1)

Where r denotes the radial distance from the Earth center in units of km, a = 6371.2 km denotes
the magnetic reference spherical radius, close to the mean value of the Earth radius, φ denotes
longitude and θ denotes geocentric latitude (i.e. 90 − latitude). When converting between
geocentric and geodetic coordinates, we ususally set major axis A and a reciprocal flattening 1/f
to be 6378.137 km and 298.257223563, respectively. P
m
n
cos(θ) denotes the Schmidt semi- (or

10
quasi-) normalized associated Legendre functions with degree n and order m. we choose the
maximum of spherical harmonic degree of the expansion N so that we can reliably determine the
coefficients of the model given the available coverage and quality of observations. And for
IGRF-11, we choose N to be 10 up to and including epoch 1995, and it is extended to degree 13
to do with the excellent data provided by the Ørsted and CHAMP satellites.
In this IGRF model, Gauss coefficients g
m
n
cos(θ) and h
m
n
cos(θ) are given for the main field (MF)
at epochs which is separated by 5-yr intervals between 1900 and 2010 A.D. Then, using the
following linear expression specifies the time-dependence of the Gauss coefficients:

Where t denotes the time of interest (in units of years) and T denotes the epoch preceding t which
must be an exact multiple of 5-yr, such that T
0
≤ t < (T
0
+ 5.0). And it is similarly for h
m
n
. The
coefficients g
.m
n
(T
0
), given in units of nTyr
− 1
, is the average first time derivative of g
m
n
(t) during
this period T
0
to T
0
+ 5.0, i.e. the linear secular variation (SV) during this period. If MF models
exist for both T
0
and T
0
+ 5.0, then we can simply calculate g
.m
n
(T
0
) by using linear interpolation
as
g
m
n
(T
0
+ 5.0) − g
m
n
(T
0
)
5.0
. For the final 5-yr of the model validity, which is the period between
2010 and 2015 for IGRF-11, the coefficients of predictive average SV g
.m
n
(T
0
) and h
.
m
n
(T
0
) will be
explicitly provided.
g
m
n
(t) = g
m
n
(T
0
) + g
.m
n
(t − T
0
) .
(2)

11
By using eq. (1), we can obtain the geocentric components of the geomagnetic field for the
northward, eastward and vertical directions (X, Y, Z) by taking appropriate components of ∇V in
spherical polar coordinates and from the coefficients in the model,
X' =
2
r

∂V
∂θ
, Y' =
1
rsin(θ)

∂V
∂ϕ
, Z' =
∂V
∂τ
.

(3)

It is often necessary to work on geodetic coordinates and to utilize the World Geodetic System
1984 datum defined above. Equations (1)–(4) of Hulot et al. (2007) describe the transformations
from geocentric to geodetic coordinates and from the geocentric field components (X’, Y’, Z’)
into the geodetic field components (X, Y, Z). Usually, the declination D, inclination I, the
horizontal field H and the total field F are required for applications; these are obtained from X, Y
and Z by using the following relationship,
H = X
2
+ Y
2
, F = X
2
+ Y
2
+ Z
2
, D = arctan
⎝
⎜
⎛
⎠
⎟
⎞
Y
X
, I = arctan
⎝
⎜
⎛
⎠
⎟
⎞
Z
H
.

(4)

Then, we will have the definition of the Geomagnetic Equator as follows,
Definition: The curve on a constant altitude spherical surface where the vertical component of
geomagnetic field is zero is called the geomagnetic equator.
The following figure illustrate the definition of geomagnetic equator,

12

Figure 1: illustration of the definition of geomagnetic equator

And by recording the longitude and latitude of each point along the geomagnetic equator in April
2011, we will get the following figure,
−1
−0.5
0
0.5
1
x 10
4
−1
0
1
x 10
4
−8000
−6000
−4000
−2000
0
2000
4000
6000
8000

geomagnetic equator
earth center
geographic equtor
earth

13

Figure 2: location of geomagnetic equator as of April 2011

In this thesis, we fix the altitude to be 400 km. Then, we can have the definition of meridional
line as follows,
Definition: Given a geomagnetic equator point, the curve on which the horizontal geomagnetic
directions of all the points are the same as that of this geomagnetic equator is called meridional
line of this geomagnetic equator.
The following figure illustrate the definition of meridional line,
−150 −100 −50 0 50 100 150
−80
−60
−40
−20
0
20
40
60
80
longitude
latitude

geomagnetic equator

14

Figure 3: illustration of the definition of meidional line

And by recording the longitude and latitude of each point along the meridional line at longitude
= -120 in April 2011, we will get the following figure,

−1 −0.5 0 0.5 1
x 10
4
−1
0
1
x 10
4 −8000
−6000
−4000
−2000
0
2000
4000
6000
8000

meridional line
geomagnetic equator
earth
geographic equtor
earth center

15

Figure 4: location of the meridional line at longitude = -120 as of April 2011

Here, we use iterative method to get the meridional line of a given geomagnetic equator point as
follows:
We know, to every point in the spherical surface with 400 km altitude component field, we have
the following equation,
B = B
h
+ B
v

(5)

−150 −100 −50 0 50 100 150
−80
−60
−40
−20
0
20
40
60
80
longitude
latitude

meridional line
geomagnetic equator

16
Here, B
h
is the horizontal geomagnetic field of this point, and B
v
is the vertical geomagnetic field
of this point.
And in each step of the iterative method, we need to set one start point v
k
. Then we move Δd
(usually Δd = arc-length of one latitude degree) along the horizontal geomagnetic direction of
the start point, and we will get a new point v'
k+1
, and we have the following equation,
v'
k+1
= v
k
+ Δd 
B
h
B
h

(6)

Then, we draw a straight line through this new point and the earth center, and get an intersection
between the straight line and the spherical surface with 400 km altitude component field. Then
we set the intersection as the start point v
k+1
. The following equation illustrate the relationship,
v
k+1
= (a+alt)
v'
k+1
v'
k+1

(7)

Where a = 6371.2 km denotes the magnetic reference spherical radius, close to the mean value of
the Earth radius, and alt is the altitude of the spherical surface, here, we set it to be 400 km.
Here, we first set this given geomagnetic equator point as the first start point, and then repeat the
process until we get all the northward points. And the following figure illustrates one step of the
iterative method,

17

Figure 5: illustration of one step of the iterative method
As to the southward points, we use the opposite direction of the horizontal geomagnetic direction
in each step. Then we will get all the southward points we need.
And using the northward and southward points and equator points, we interpolate them by using
spline. Then, we get the meridional line for this geomagnetic equator point.
We can get the VTEC data of some positions on the earth from the GAIM++ Model at 00:00 UT,
April 26
th
2011. By using ‘spline’ method, we can interpolate the data to get the relationship
between the position and its corresponding VTEC. Then we input the longitude and latitude of
−1
−0.5
0
0.5
1
x 10
4
−1
−0.5
0
0.5
1
x 10
4
−8000
−6000
−4000
−2000
0
2000
4000
6000
8000

earth center
START POINT
INTERSECTION

18
the points along some meridional line, and then get the VTEC map in this meridional line. The
following figure shows the VTEC map along the meridional line at longitude = -120.

Figure 6: VTEC map along the meridional line at longitude = -120 as of 2011/04/26 00:00 UT

By viewing the VTEC map along a geomagnetic meridional line, we can find that there is at least
one peak in both the right and left sides of the equator position in each meridional line, then we
define the longitude corresponding the nearest right peak as right latitude, denoted as latL.
Similarly, we define the latitude corresponding the nearest left peak as left latitude, denoted as
latR.
−80 −60 −40 −20 0 20 40 60 80
0
10
20
30
40
50
60
Latitude
VTEC
Longitude= −120

VTEC
lat0

19
If there exist right latitude and left latitude in the VTEC map along one meridional line, we can
have the definition of Equatorial Anomaly Gap (EAP) at this meridional line as follows,
Definition: The difference between right latitude and left latitude in the VTEC map at one
meridional line is called as Equatorial Anomaly Gap at this meridional line.
The following figure illustrates the definition of EAP,

Figure 7: definition of EAP in meridional line at longitude = -120 as of 2011/04/26 00:00 UT

And if there isn’t right latitude, left latitude or both in the VTEC map along one meridional line,
then we set EAP in this meridional line to be 0.
−80 −60 −40 −20 0 20 40 60 80
0
10
20
30
40
50
60
Latitude
VTEC
Longitude= −120

VTEC
lat0
latL
latR
EAP

20
However, if the shape of the VTEC map at some meridional line is too flat, it will be hard to
detect the right latitude and left latitude. So we have the following empirical approach to
estimate the EAP at some given meridional line:
First, we draw a straight line, called horizontal line, parallel to the latitude axis, and check
whether this line will intersect the VTEC map. If there are at least two intersection whose
latitudes are respectively larger than and less than that of the equator position, then we denote the
least latitude whose latitude is larger than that of the equator position as lat1 and the largest
latitude whose latitude is less than that of the equator position as lat2.
Then we denote the latitude of the equator position as lat0. And we illustrate those positions in
the following figure,

Figure 8: some positions of estimation method in meridional line at longitude = -120 as of
2011/04/26 00:00 UT
−100 −80 −60 −40 −20 0 20 40 60 80 100
0
10
20
30
40
50
60
Latitude
VTEC
longitude = −120

VTEC
horizonal line
lat0
lat2
lat1

21
Given the above positions of some given meridional line, we can use an empirical approach to
estimate the EAP at this meridional line, and have the following equation:
EAP =
⌡
⌠
lat1
lat0
VTEC(x)xdx

⌡
⌠
lat1
lat0
VTEC(x)dx
−
⌡
⌠
lat0
lat2
VTEC(x)xdx

⌡
⌠
lat0
lat2
VTEC(x)dx

(8)

22
2.2 Sensitivity and Validation study
In this section, we will present the sensitivity of our model. We know there are two parameters,
the 10.7cm solar radio flux (F10.7) and the planetary geomagnetic activity index (AP),
influencing the VTEC map, and then influencing the EAP value. Therefore, we change the value
of F10.7 and AP to get the relationship between them and EAP value.
First, using the different F10.7, we change the value of AP and then get the dependency between
AP and the maximum of the EAP by estimation. The following figure shows this dependency,

Figure 9: maximum of EAP by estimation vs AP as of 2011/04/26 00:00 UT
5 10 15 20
22.25
22.3
22.35
22.4
22.45
22.5
22.55
22.6
ap
maxeap by estimation
f10.7=100
5 10 15 20
26
26.5
27
27.5
28
28.5
ap
maxeap by estimation
f10.7=150
5 10 15 20
32
32.5
33
33.5
34
34.5
35
ap
maxeap by estimation
f10.7=200
5 10 15 20
34.6
34.7
34.8
34.9
35
35.1
35.2
35.3
ap
maxeap by estimation
f10.7=250

23
Then, by using the different AP, we change the value of F10.7 and then get the dependency
between F10.7 and the maximum of the EAP by estimation. And the following figure shows this
dependency,

Figure 10: maximum of EAP by estimation vs F10.7 as of 2011/04/26 00:00 UT

100 150 200 250
22
24
26
28
30
32
34
36
f10.7
maxeap by estimation
ap=5
100 150 200 250
22
24
26
28
30
32
34
36
f10.7
maxeap by estimation
ap=10
100 150 200 250
22
24
26
28
30
32
34
36
f10.7
maxeap by estimation
ap=15
100 150 200 250
22
24
26
28
30
32
34
36
f10.7
maxeap by estimation
ap=20

24
2.3 Correlation of ExB velocity and EAP
In this section, we first determine how the two parameters, F10.7 and AP, influence the ExB
velocity by the following techniques.
First, by using the different F10.7, we change the value of AP and then get the dependency
between AP and the maximum of the ExB velocity at 00:00 UT, April 26
th
2011. And the
following figure shows this dependency,

Figure 11: maximum of ExB velocity vs AP as of 2011/04/26 00:00 UT

25
Then, by using the different AP, we change the value of F10.7 and then get the dependency
between F10.7 and the mximum of the ExB velocity at 00:00 UT, April 26
th
2011. And the
following figure shows this dependency,

Figure 12: maximum of ExB velocity vs F10.7 as of 2011/04/26 00:00 UT

26
Now, we can combine the above results to determine the dependency between the Estimation
EAP in the Earth’s ionosphere and the ExB velocity of the Earth’s ionosphere.
By fixing AP, we can get the dependency between the maximum of EAP by estimation and the
maximum of the ExB velocity at 00:00 UT, April 26
th
2011 in the following figure,

Figure 13: maximum of Estimation EAP vs maximum of ExB velocity by fixing AP as of
2011/04/26 00:00 UT
3500 3550 3600 3650 3700 3750
22
24
26
28
30
32
34
36
maxexb
maxeap by estimation
ap=5
3500 3550 3600 3650 3700 3750
22
24
26
28
30
32
34
36
maxexb
maxeap by estimation
ap=10
3500 3550 3600 3650 3700 3750
22
24
26
28
30
32
34
36
maxexb
maxeap by estimation
ap=15
3500 3550 3600 3650 3700 3750
22
24
26
28
30
32
34
36
maxexb
maxeap by estimation
ap=20

27
And by fixing F10.7, we can get the dependency between the maximum of EAP by estimation
and the maximum of the ExB velocity at 00:00 UT, April 26
th
2011in the following figure,

Figure 14: maximum of Estimation EAP vs maximum of ExB velocity by fixing F10.7 as of
2011/04/26 00:00 UT

3540 3540.5 3541 3541.5 3542 3542.5
22.25
22.3
22.35
22.4
22.45
22.5
22.55
22.6
maxexb
maxeap by estimation
f10.7=100
3601 3601.5 3602 3602.5 3603 3603.5
26
26.5
27
27.5
28
28.5
maxexb
maxeap by estimation
f10.7=150
3662.5 3663 3663.5 3664 3664.5 3665
32
32.5
33
33.5
34
34.5
35
maxexb
maxeap by estimation
f10.7=200
3699.5 3700 3700.5 3701 3701.5 3702
34.6
34.7
34.8
34.9
35
35.1
35.2
35.3
maxexb
maxeap by estimation
f10.7=250

28
Chapter 3: Conclusion
As a continuation of the efforts in monitoring and forecasting the conditions of the Earth’s
ionosphere, the work presented in this thesis shows the dependency between the EAP in the
Earth’s ionosphere and the ExB velocity of the Earth’s ionosphere. And, we have examined how
the main parameters, the 10.7cm solar radio flux (F10.7) and the planetary geomagnetic activity
index (AP) influence the two characters of the Earth’s ionosphere.
Our results have shown that EAP is positively correlated with F10.7 whenever AP is
small or large. In addition, whenever F10.7 is large or small, we cannot clarify the dependency
between the EAP and AP.
As to ExB velocity, we can conclude that AP cannot influence ExB velocity whatever
F10.7 we choose. But as to F10.7, ExB velocity is positively correlated with it whenever AP is
small or large.
At last, it is clear that EAP is positively correlated with ExB velocity whenever AP is
small or large. And compared with the estimation of ExB velocity from an OSSE, the program in
this thesis is in an actual use with real data, so there are no independent measurements of the
driving forces. And this program cannot determine whether or not the estimated driving force,
ExB velocity, is more accurate than the climatological value. Considerable experience with the
combination between GAIM++ and IGRF must be gained to calibrate and modify the program to
produce reliable forecast.

29
References
[1] Akopian, V. (2008), Modeling of earth's ionosphere and variational approach for data
assimilation (Doctoral dissertation). Retrieved from usc.edu/library.
[2] G. Hajj, B.Wilson, C.Wang, X. Pi, and G. I. Rosen (2004), Data assimilation of ground gps
total electron content into a physics-based ionospheric model by use of the kalman filter, Radio
Science 39.
[3] Hogg, M. A., & Reid, S. A. (2006). International Geomagnetic Reference Field: the eleventh
generation. Geophysical Journal International, 16, 7-30. doi:10.1111/j.1365-246X.2010.04804.x

[4] Hulot, G., Olsen, N. & Sabaka, T.J., (2007). The present field, geomagnetism,
in Treatise on Geophysics, Vol. 5, pp. 33–75, ed. Schubert, G., Elsevier, Amsterdam.

[5] Iijima, B. A., I. L. Harris, C. M. Ho, U. J. Lindqwister, A. J. Mannucci, X. Pi, M. J. Reys, L.
C. Sparks, and B. D. Wilson. (1999), Automated daily process for global ionospheric total
electron content maps and satellite ocean altimeter ionospheric calibration on global positioning
sys data, J. Atmos. Sol. Terr. Phys., 61, 1205.

[6] Mannucci, A. J., B. D. Wilson, D. N. Yuan, C. M. Ho, U. J. Lindqwister, and T. F. Runge.
(1998), A global mapping technique for GPS-derived ionospheric total electron content
measurements, Radio Sci., 33(3), 565.

[7] MathWorks, Matlab, Retrieved from http://www.mathworks.com/.
[8] Pi, X., M. Mendillo, M.W. Fox, and D. N. Anderson. (1993), Diurnal double maxima
patterns in the F region ionosphere: Substorm-related aspects, J. Geophys. Res., 98(13), 677.
[9] Pi, X., Wang, C., Hajj, G.A., Rosen, G., Wilson, B.D., and Bailey, G.J. (2003), Estimation of
eb drift using a global assimilative ionospheric model: An observation system simulation
experiment, Journal Geophysical Research, 108, A2, 1075, doi: 10.1029/2001JA009235.

Abstract (if available)

Abstract For the development of wireless communication, the Earth's ionosphere is very critical. A Matlab program is designed to improve the techniques for monitoring and forecasting the conditions of the Earth's ionosphere. The work in this thesis aims to modeling of the dependency between the equatorial anomaly gap (EAP) in the Earth's ionosphere and the crucial driver, ExB velocity, of the Earth's ionosphere. ❧ In this thesis, we review the mathematics of the model in the eleventh generation of the International Geomagnetic Reference Field (IGRF) and an enhancement version of Global Assimilative Ionospheric Model (GAIM), GAIM++ Model. We then use the IGRF Model and a Vertical Total Electron Content (VTEC) map from GAIM++ Model to determine the EAP in the Earth's ionosphere. Then, by changing the main parameters, the 10.7cm solar radio flux (F10.7) and the planetary geomagnetic activity index (AP), we compare the different value of the EAP in the Earth's ionosphere and the ExB velocity of the Earth's ionosphere. At last, we demonstrate that the program can be effective in determining the dependency between the EAP in the Earth's ionosphere and the ExB velocity of the Earth's ionosphere.

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Modeling of Earth's ionosphere and variational approach for data assimilation

PDF

Determining blood alcohol concentration from transdermal alcohol data: calibrating a mathematical model using a drinking diary

PDF

A nonlinear pharmacokinetic model used in calibrating a transdermal alcohol transport concentration biosensor data analysis software

PDF

Identifying important microRNAs in progression of breast cancer

PDF

Improvement of binomial trees model and Black-Scholes model in option pricing

PDF

An FDTD model for low and high lightning generated electromagnetic fields

PDF

An abstract hyperbolic population model for the transdermal transport of ethanol in humans: estimating the distribution of random parameters and the deconvolution of breath alcohol concentration

PDF

Automated classification algorithms for high-dimensional data

PDF

A Bayesian approach for estimating breath from transdermal alcohol concentration

PDF

Homologies, chain complexes, and the shape of data

PDF

Obtaining breath alcohol concentration from transdermal alcohol concentration using Bayesian approaches

PDF

Linear filtering and estimation in conditionally Gaussian multi-channel models

PDF

Estimation of random input to semi-linear abstract parabolic systems with application to quantitative description of drinking behavior based on trans-dermal alcohol concentration

PDF

Distributed parameter model based system identification and filtering in the estimation of blood alcohol concentration from transdermal alcohol biosensor data

PDF

Supervised learning algorithms on factors impacting retweet

PDF

Tamed and truncated numerical methods for stochastic differential equations

PDF

Bulk and edge asymptotics in the GUE

PDF

Automatic tracking of protein vesicles

PDF

Asset price dynamics simulation and trading strategy

PDF

A fully discrete approach for estimating local volatility in a generalized Black-Scholes setting

Asset Metadata

Creator Ao, Xi (author)

Core Title Empirical approach for estimating the ExB velocity from VTEC map

School College of Letters, Arts and Sciences

Degree Master of Science

Degree Program Applied Mathematics

Publication Date 07/28/2014

Defense Date 06/18/2014

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag ExB velocity,OAI-PMH Harvest

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Wang, Chunming (committee chair), Lototsky, Sergey V. (committee member), Rosen, I. Gary (committee member)

Creator Email albertdream1991@hotmail.com,xao@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-448978

Unique identifier UC11286636

Identifier etd-AoXi-2747.pdf (filename),usctheses-c3-448978 (legacy record id)

Legacy Identifier etd-AoXi-2747.pdf

Dmrecord 448978

Document Type Thesis

Format application/pdf (imt)

Rights Ao, Xi

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA