2017 FULL SOLAR ECLIPSE: OBSERVATONS AND LWPC MODELING OF VERY LOW
FREQUENCY ELECTROMAGNETIC WAVE PROPAGATION
JAMES R. BITTLE B.S., Texas Christian University, 2016
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering
JAMES R. BITTLE ALL RIGHTS RESERVED
This thesis for the Master of Science degree by James R. Bittle Has been approved for the Electrical Engineering Program by
Mark Golkowski, Chair Stephen Gedney Vijay Harid
Bittle, James R. (M.S., Electrical Engineering)
2017 Full Solar Eclipse: Observations and LWPC Modeling of Very Low Frequency Electromagnetic Wave Propagation
Thesis directed by Associate Professor Mark Golkowski
On August 21, 2017 a total solar eclipse occurred over the United States commencing on the west coast moving across to the east coast providing an opportunity to observe how the rapid day-night-day transition changed the ionosphereâ€™s D-region electron density and how very low frequency (VLF) electromagnetic wave propagation was affected. To observe the solar obscurity effects, VLF receivers were deployed in two locations: one in the path of totality in Lakeside, Nebraska and another south of the totality path in Hugo, Colorado. The locations were chosen to achieve an orthogonal geometry between the eclipse path and propagation path of U. S. Navy VLF transmitter in North Dakota, which operates at 25.2 kHz and has call sign NML. VLF amplitude and phase changes were observed in both Lakeside and Hugo during the eclipse. A negative phase change was observed at both receivers as solar obscuration progressively increased. The observed phase changes became positive as solar obscuration reduced. The opposite trend was observed for the amplitude of the transmitted signal: growth as max totality approached and decay during the shadowâ€™s recession. The Long Wave Propagation Capability (LWPC) code developed by the US Navy was used to model the observations. LWPC is a modal solution finder for Earth-ionosphere waveguide propagation that takes into account the D-region density profile. In contrast to past efforts where a single ionosphere profile was assumed over the entire propagation path, a degree of spatial resolution along the path was sought here by
solving for multiple segments of length 100-200 km along the path. LWPC modeling suggests that the effective reflection height changed from 71 km in the absence of the eclipse, to 78 km at the center of the path of totality during the total solar eclipse and is on agreement with past work.
The form and content of this abstract are approved. I recommend its publication
Approved: Mark Golkowski
To my wife who would always match my enthusiasm (or pull me out of distress) relating to my studies, to my parents who provided me with wisdom that cannot be gained without experience in the same matter, and to my friends who always believe in me, thank you. Any success I have is due to the accumulation of love and support I feel from you all. I am truly blessed
Thank you to Dr. Weis at Texas Christian University for always pushing me to understand matters at a level below the surface and thank you Dr. Golkowski for the opportunities you gave me to develop as a young scientist and engineer. The guidance, patience and tolerance you provided to me is one-of-a-kind and I will cherish our work together for the rest of my life. This work was supported by the National Science Foundation under awards AGS-1451210, and OPP-1542608 to University of Colorado Denver.
TABLE OF CONTENTS
1.1 Plasma and the Ionosphere..........................................1
1.2 The Earth - Ionosphere Waveguide...................................5
1.3 VLF Wave Sensing and Measurements..................................6
1.4 LWPC Modeling......................................................7
1.4.1 LWPC Configuration....................../...................7
1.4.2 Past Work of LWPC Modeling Solar Eclipse Conditions.........10
II. EQUIPMENT DEPLOYMENT.......................................................13
2.1 Receiver Geometry in Relation to NLM Transmitter and Eclipse Path.13
2.2 Receiver Equipment................................................15
2.2.1 Loop Antennas...............................................15
2.2.2 Pre-amplifier and Line Receiver.............................17
2.2.3 Postprocessing and OS.......................................18
3.1 Antenna Rotation..................................................20
3.2 Hugo, Colorado....................................................21
3.3 Lakeside, Nebraska................................................23
3.4 Solar Flare.......................................................25
IV. LWPC MODEL.................................................................28
4.1 Modeling the Horizontal Magnetic Field Component in LWPC..........28
4.2 Propagation Path Segmentation.....................................29
4.3 /i â€™ and P Step Gradi ents........................................31
4.4 Global Error Calculations.........................................32
4.5 LWPC Solution.....................................................40
V. SUMMARY AM) CONCLUSIONS....................................................42
LIST OF TABLES
1.4.1 Example of input parameters to LWPC code.........................................8
1.4.2 Example h â€™ and Pâ€™s for each segment of the generated waveguide..................8
1.5 Summary of eclipse effects on amplitude and phase of signals received ........12
LIST OF FIGURES
1.1 Ionospheric electron density as a function of height for day and night conditions.....3
1.2 Modeled ionospheric electron density..................................................4
1.3 Modeled electron density over measured electron density in the D - Region of the
1.4.1 Illustrated Example h â€™ and for each segment of the generated waveguide...............8
1.4.2 A map of the paths from European transmitters to receiver sites around Europe ....11
2.1 Receiver locations in relation to NLM transmitter and totality path. The red line
represents the eclipse shadow on the ground while the blue line represents the eclipse shadow on the ionosphere............................................................13
2.2 Shadow on ionosphere compared to shadow on ground due to angle of sun................14
2.3 Right isosceles triangle loop antennas...............................................16
2.4 Pre - amplifier......................................................................17
2.6 Receiver packages....................................................................19
3.1 Hugo, CO: Southern Site..............................................................21
3.2 Observed amplitude and phase at Hugo, CO.............................................22
3.3 Lakeside, NE: Totality Site..........................................................23
3.4 Observed amplitude and phase at Lakeside, NE.........................................24
3.5 Observed C - class solar flare at end of max obscuration at Lakeside, NE ........25
3.6 Both sites compared to solar flare and solar obscuration at Lakeside, NE.............26
4.2 Zeroth order ionosphere shape compared to imagined shape over the path of
4.3 Illustration of different h â€™ and P step-gradients per segment. Note that the Lakeside, NE
receiver is at the end of boundary three (579 km from the NLM transmitter) and the receiver at Hugo, CO is at the end of boundary five (930 km from the NLM transmitter)........................................................................31
4.4 Three global error methods compared to different step-gradient types at tau = 3
4.5 Three global error methods compared to different step-gradient types at tau = 10
4.6 Three global error methods compared to different step-gradient types at tau = 18
4.7 Comparison of global error for different time steps before and after totality......39
4.8 Observed data vs. LWPC model.......................................................40
1.1 Plasma and the Ionosphere
A Greek philosopher by the name of Empedocles believed that all matter was a ratio of four elements: earth, water, air, and fire , A modern understanding of physics shows that the notion of four elements was strikingly on target; just limited in scope compared to the modern definition of the four phases of matter: solid, liquid, gas, and plasma. In fact, fire is a form of plasma, though not the only type. Plasma is simply any gas with a high enough temperature or energy level that the electrons are stripped away from their parent molecules. The conditions necessary for creating plasma are actually common on the scale of the universe. In fact, 99% of the known universe is in a plasma-state and the fact planet Earthâ€™s surface is a unique exception to this is one of the reasons that humans and numerous other species thrive here. But our little bubble of non-plasma peace isnâ€™t as definitive as one might think.
At a height of about 60 km above the earthâ€™s surface, the Ionosphere begins to take shape. The ionosphere is a combination of neutral particles, ions, and free electrons that exists in a range of altitudes 60-1000 km above sea level. At these altitudes solar radiation (photons) and even extrasolar cosmic events bombard neutral gaseous particles within our atmosphere and cause separation of electrons thus forming a plasma state. The ionosphere is divided into D, E, and F regions, D being the lowest (60-95 km) and having the lowest density of free electrons. The reason for a lower concentration of electrons in the D region can be understood using the zeroth order Boltzmann equation under an assumption of steady state.
At steady state the left side of (1.1) representing time and spatial variations in the electron density equates to zero while â€” aN2 represents the recombination loss rate for ions and electrons, â€”vaN is the attachment loss rate of electrons to neutrals, v,/V is the collision rate of free electrons knocking other electrons free, and Q is the photoionization rate, or the rate at which photons knock electrons away from their host particle. All four of these terms are heavily dependent on altitude and day vs. night conditions. For daytime Q is nonzero and increases with altitude since the number of solar photons capable of ionizing neutrals decreases closer to the Earthâ€™s surface. At very high altitude above 400 km, the solar radiation is high but there is little atmosphere left to ionize. The ionosphere thus has a maximum electron density in between 200km and 400 km altitude. Figure 1.1 displays how the electron densities behave logarithmically as a function of height for given day and night profiles.
DAY/NIGHTTIME ELECTRON CONCENTRATIONS
Figure 1.1: Ionospheric electron density as a function of height for day and night
In the E and F regions (100 km - 500 km), there are sufficient densities of electrons allowing incoherent scatter radars (ISR) to directly measure the density from earth. Spacecraft can also pass through regions of the ionosphere as low as 300 km. But, at altitudes of 60-95 km, the density of electrons is just too low to measure with standard ISR techniques, too high for weather balloons, and too low for satellite measurements. Direct D-region diagnostics have to be done with costly high-altitude rockets. Based on these measurements, a still widely accepted
model of the D - region electron densities at night time and day time was developed by Waits and Spies in 1964  where electron density as a function of altitude, /?, is given as
ne = n0e Â°-15ft,e(/? o.i5)(h 3]
where n0 = 1.43 x 107 [electrons cm"3], h â€™ (km) describes the effective reflection height for VLF waves and , (5 (km"1) is the â€˜sharpnessâ€™ of the profile. This model is used to generate typical day and night profiles as shown in Figure 1.2.
DAY/NIGHTTIME ELECTRON CONCENTRATIONS
Figure 1.2: Modeled ionospheric electron density Figure 1.3 shows how the D region profile from Equation (1.2) overlaps the full profile
for altitudes below 100 km.
Figure 1.3: Modeled electron density over measured electron density in the D -
Region of the Ionosphere
1.2 The Earth - Ionosphere Waveguide
The plasma within the ionosphere has its own â€˜frequencyâ€™ or in other words, an
oscillation that the ions or electrons will experience due to their density. Specifically, the plasma frequency in Hz is equal to 9(N)m where Nis the free electron density in units of m"3. The plasma frequency for the ionosphere can vary on the order of 1 - 50 MHz. For high
frequency/small-wave signals like GPS and satellite communications where the wave frequency is much higher than the plasma frequency, the electromagnetic radiation passes through the plasma without any problem. But, for AM radio waves around 1 MHz used for radio broadcast, the ionosphere is opaque and reflects those signals back down to the surface of the earth. More important to this study is the ionosphereâ€™s effect of Very Low Frequency (VLF) waves. Because VLF waves are in the range of 1-40 kHz, the wavelengths are so long that they reflect at the lowest level of the ionosphere (D region). VLF waves reflect effectively not only from the ionosphere, but also from the earthâ€™s crust which provides another reflective surface. The D region and surface of the earth ultimately create an earth-ionosphere waveguide for VLF waves which can propagate distances on the order of the earthâ€™s circumference with minimal attenuation. While the earthâ€™s crust is a stable conducting surface, the ionosphere, especially the D region, is very dynamic due to the day and night transitions. For that reason, a solar eclipse provides a rare opportunity to observe how the rapid day-night-day transition effects the propagation of VLF waves.
1.3 VLF Wave Sensing and Measurements
Because of the natural waveguide between the earth and ionosphere we can observe a signal from a known source such as Navy VLF transmitters and measure how the amplitude and phase of that signal changes. The measured amplitude and phase at a receiver is the end result of multimodal propagation within the earth-ionosphere waveguide. The conditions of the waveguide such as its reflection height and relative conductivity at said height are directly related to equation 1.1, thus transmitted VLF waves from a known source power provides a means to indirectly dictate the electron density of the D - region ionosphere which can then be used to deduce other sets of valuable information relating to the ionospheric conditions. Besides
man made VLF signals, natural VLF emissions from lightning (consider the current from lightning strike as approximated by a delta function in time, thus radiating nearly infinite bandwidth in the frequency domain) provide another diagnostic tool.
1.4 LWPC Modeling
In the 70â€™s to early 80â€™s of the 20th century, the US military was heavily invested in using VLF transmitters not only for communicating to submerged vessels by exploiting VLF wavesâ€™ ability to penetrate deep into sea water, but also for global positioning. When trying to send a VLF signal to a specific location, or rather receive a signal providing the location of a desired target, the ionosphereâ€™s dynamic nature would change the propagation channel thus changing the amplitude and phase of a signal, creating error. In order to model these amplitude and phase changes, the Long Wave Propagation Capability code was developed by the US Navy which utilized mode theory in a two-dimensional waveguide geometry to help determine an expected amplitude and phase at a receiving point from a transmitter at a known location.
1.4.1 LWPC Configuration
The LWPC code was written in Fortran. At the user level of the code, there are nine main inputs: Transmitter power (1), transmitter frequency (2), transmitter latitude (3), transmitter longitude (4), receiver latitude, (5), receiver longitude (6), boundaries (7), segments (8), and electron density (9). The electron density is determined by using (1.1). So, at a level below the surface of the codeâ€™s inner workings, a user chooses an h â€™ and value, which are used to calculate the electron density for a single segment and boundary. You can segment the waveguide up to ten times with unique h â€™ and /f s thus providing 10 different electron densities, waveguide segment heights, and conductivities of said waveguide. Tables 1.4.1 and 1.4.2
demonstrate the input parameters for the LWPC code and Figure 1.4 depicts visually how the
code would create a waveguide for â€˜five â€˜segments.
Table 1.4.1: Example of input parameters to LWPC code
Tx Power (kW)Tx Freq (kHz) Tx Lat Tx Long Rx Long Rx Lat Boundaries (km) Segments Ne
500 25.2 46.366 98.3356 42.0481 102.4075 100, 200, 300, 400, 500 5 determined by table 1.4.2
Table 1.4.2: Example h â€™ and Pâ€™s for each segment of the generated waveguide
Segment 1 2 3 4 5
H prime (km) 70 71 72 71 70
Beta (kmA-l) 0.4 0.42 0.44 0.42 0.4
1 75 b
H prime Values for Example Segments
(3 Values for Example Segments
â€” 0.45 b
0 100 200 300 400 500
Figure 1.4.1: Illustrated example of h â€™ and Pâ€™s for each segment of the generated waveguide
Once these input parameters are set the LWPC follows these three steps:
1. PRESEG - In order to properly create a model of an earth ionosphere waveguide the earthâ€™s magnetic field and conductivity/permittivity must be known at the location of the waveguide , The PRESEG code takes the user - input information above congruent to
the excepted model of the earths conditions at the given location to then separate the Tx-Rx path into the desired sections creating a homogeneous profile per section and then summing the sections for one large waveguide with different parameters intermediately.
2. MODEFNDR - All modes of propagation must be pre-calculated since the received amplitude and phase will be a sum of all such modes. The waveguide parameters determined in PRESEG are passed into MODEFNDR so that the modes of propagation and excitation factors can be determined. Then, MODEFNDER finds the Eigenangles for each mode of propagation by determining which angles of incidence satisfy the mode condition for propagation:
det[/ â€” Rl(Q)Ru(Q)] = 0,
â– ||/?ll(0) 0 and Ru = ' 11*11(9) ||*i(0)'
0 nWJ .i*l|(0) i*i(0).
Each of the Rl and Ru tensors specify the reflection coefficients for the lower and upper parts of the waveguide. The lower reflection coefficient (Rl) is dependent on properties of the earthâ€™s crust (permittivity, conductivity, and angle of incidence) where isotropic properties are an acceptable approximation, hence the off-diagonal terms are zero. The upper reflection coefficient (Ru) represents an anisotropic environment because the ionospheric plasma is magnetized by the earthâ€™s magnetic field, thus the polarization of the VLF wave will determine how the VLF wave reflects down to the earthâ€™s crust. Further detail on this derivation can be reviewed in ,
3. FASTMC - Once all the modes of propagation are calculated, boundary conditions at each discontinuity within the segmented waveguide must be accounted for. This is done by calculating height gains, excitation coefficients, and cumulative mode conversion coefficients  in a waterfall at each segment until the receiver is reached. At this point, the sum of the modes that made it to the receiver are calculated and an amplitude is given in dB with respect to 1 pV/m as well as a total phase change over the whole path given in degrees.
1.4.2 Past work on LWPC Modeling of Solar Eclipse Conditions
Researchers have attempted to analyze the effects of a solar eclipse on the ionosphere in the past [9, 12], Clilverd, et al  was able to look at many different transmitter -receiver paths that intersected the eclipse path of totality for a 1999 eclipse in Europe and organized the observations into short (<1,000 km), medium (1,000-10,000km), and long (>10,000 km) VLF propagation paths. This experiment setup can be seen in Figure 1.4.2 and Table 1.5. Within the short paths described above, amplitude changes of +2.3 to +8.5 dB were observed, but lacked data on the phase changes. Notice that many of the short paths intercept the path of totality, but almost all angle of incidences from the transmitter on the path of totality are oblique. Due to this, all the observed amplitudes and phases were different because each VLF wave encountered a different form of the ionosphere as it changed rapidly during the eclipse obscuration.
With many variables to model, Clilverd et al.  chose to represent the ionosphere as one homogeneous profile so that a single electron density profile was seen over the entire path from a transmitter to a receiver. This was necessary for a reasonable model but due to this fact, greater averaging and approximations were made to best match measured amplitude and phase of
of the ionospheric profile were
observed data and spatial gradients
in the actual nature
Table 1.5: Summary of eclipse effects on amplitude and phase of signals received
Transmitter-Receiver Frequency, kHz Distance, km Maximum Obscuration, % Amplitude, dB Phase, deg
GBR Cambridge 16.0 90 95 +0.7 n/a
GBZ-Cambridge 19.6 220 95 -3.0 n/a
FTA2-Cambridge 20.9 440 100 -1.3 n/a
D H 0â€”C ambr idge 23.4 520 95 +0.5 n/a
FTA2-Saint Ives 20.9 610 100 +8.5 -70
DHO-Saint Wolfgang 23.4 730 100 +5.9 n/a
FTA2-Saint Wolfgang 20.9 810 100 +2.3 n/a
DHO-Budapest 23.4 1,060 99 +3.0 n/a
GBR-Saint Wolfgang 16.0 1,160 100 +0.7 n/a
FTA2-Budapest 20.9 1,240 100 -4.0 -70
GBZ-Saint Wolfgang 19.6 1,290 100 +3.0 n/a
GBR-Budapest 16.0 1,560 99 â€”3.0/+2.5 +40
GBZâ€”Budapest 19.6 1,700 99 +4.0 -40
FTA2-Halley 20.9 13,960 88 -0.3 -5
GBZ-Halley 19.6 14,330 100 -0.7 -25
GBR-Halley 16.0 14,340 100 -1.0 n/a
DHO-Halley 23.4 14,510 100 -1.1 -29
The experiment set up described in this thesis focuses on what Clilverd el al  describes as a short path. The amplitude and phase changes to VLF signals at paths under 1000 km is analyzed and compared to modeling. Due to the dynamic nature of the ionosphere, an innovative approach was taken: Segmentation of the earth - ionosphere waveguide was implemented to better capture the true shape of the ionosphere as the eclipse shadow passed over a region. This novel approach proved to give a finer spatial resolution of how the ionosphere changed in relation to the solar obscuration percentage.
2.1 Receiver Geometry in Relation to NML Transmitter and Eclipse Path
Figure 2.1: Receiver locations in relation to NML transmitter and path of totality.
The red line represents the eclipse shadow on the ground while the blue line represents the eclipse shadow on the ionosphere
In North Dakota there is a VLF transmitter that operates at a frequency of 25.2 kHz and a transmitting power of 500 kW under the call sign of NML. Conveniently, this transmitter was near the 2017 eclipse path of totality, so close in fact that the transmitter experienced an 80% solar eclipse, locally. With a transmitter selected, the next step in the process was to select receiving locations in relation to the transmitter and path of totality. It was decided that the best solution for observing clearly related differences in amplitude and phase as well as constraining
parameters for an ionosphere model was to align receivers along a normally incident great circle path from the transmitter in relation to the path of totality. Figure 2.1 illustrates this geometry.
The precise location of each receiver was chosen based on the availability of BLM (Bureau of Land Management) land that could be accessed without restrictions, and also along the transmitter-receiver overlapping path. Geographic exactness was a challenge but was successful on an order of - 100 km. The locations where located and titled as such: Hugo, CO -Southern site (38.891326, -103.410977), Lakeside, NE - Totality site (42.04643, -102.4091), and lastly Pierre, SD - Northern site (44.312838, -100.170509). Notice that the Totality site is in the center of the ground shadow but is at the edge of totality in relation to the shadow on the ionosphere. The reason the shadows are in separate locations is to the angle of which the sun was in the sky, as shown in Figure 2.2.
Figure 2.2: Shadow on ionosphere compared to shadow on ground due to angle
Unfortunately, the morning of the eclipse a large storm producing large wind gusts, hail, and flash flooding disrupted recording at the Northern site just hours before the eclipse leaving just the Totality site and Southern site. Luckily, the positioning of the Totality site still provided rich data that we thought we would only see in the Northern site. A discussion of the results can be found in Chapter three.
2.2 Receiver Equipment
To accurately and precisely measure the amplitude and phase changes of the 25.2 kHz signal throughout the duration of the solar eclipse, the AWESOME (Atmospheric Weather Electromagnetic System for Observation, Modeling, and Education) instrument was used. This system had the required sensitivity, frequency and phase response, and timing accuracy via GPS signal. Further information on the AWESOME instrument can be found in the work by Cohen et al. , The AWESOME package consists of four main components:
1. Right - Isosceles Triangle Air-Core Loop antennas
4. Sampling card and PC for data storage
2.2.1 Loop Antennas
Twelve turns of wire were used to form a 1.69 m2 air-core loop antenna that provided an inductance of 1 mH and a resistance close to 1 Q. These antenna characteristics provide an impedance match to the receiver preamplifier electronics. , The loops rely on faradayâ€™s law
so that a changing magnetic field induces a voltage on the wire which entices an equivalent current. For VLF waves inside of the earth ionosphere waveguide where TMi mode propagation is the dominant mode , this type of antenna is best suited for collecting the energy from a VLF source. Figure 2.3(a) and 2.3(b) show a set of antennas oriented in magnetic NS and EW at Hugo, CO, and Lakeside, NE the morning of the eclipse.
Figure 2.3 (a): Antennas deployed in Hugo, CO
Figure 2.3 (b): Antennas deployed in Lakeside, NE
2.2.2 Pre-amplifier and Line Receiver
The strength of the magnetic fields being detected are on the order of 1-100 pT. This induces very small signals or the order of pV that need to be amplified. Figure 2.4 shows one of the amplifiers used during the eclipse study.
Figure 2.4: Pre-amplifier.
The next step in the signal processing is to take the amplified signal and filter it before it is converted to a digital signal, as well as applying an associated time stamp. This is done by combining GPS information with the received signals into one output that is sent to the computer for post processing and filtering. This piece can be seen in Figure 2.5
Figure 2.5: Receiver
2.2.3 Post Processing and OS
The last step is data acquisition and conversion to a file format that can be easily analyzed in Matlab. For this experiment two different types of software were used. At the Hugo, CO site, a software and hardware interface designed by The University of Florida was used. It operates using a GUI interface on a Windows XP platform. This system also utilizes an external
GPS clock which some argue provides a more stable phase reference signal than the internal disciplined oscillator of the Stanford system. The latter runs on a Windows 7 platform. The Stanford software is controlled with command line prompts that call batch files without the use of a GUI for optimal performance and provided a smaller percentage of software crashes. Instead of an external GPS, it houses a GPS disciplined crystal oscillator within the receiver which allows for less equipment and power usage. Figure 2.6 shows a flow chart of both systems for easy comparison.
Pre-amplifier | Pre-amplifier
1 l f 1
Receiver and Internal GPS Receiver
(Windows 7) (Windows XP)
Figure 2.6: Receiver Packages
3.1 Antenna Rotation
At both Hugo, CO and Lakeside, NE, two right-isosceles loop antennas were set up orthogonal to one another as seen in Figures 2.3(a) and 2.3(b) with the NS antenna aligned with magnetic-north using an analog compass. This was done because magnetic-north can be found easily at both receiving sites as opposed to trying to orient the antennas to point directly at the transmitter. In postprocessing the data from the orthogonal antennas was digitally rotated so that each set of antennas were virtually positioned with the plane of the NS antenna pointed at the transmitter. This allowed us to decompose the total horizontal magnetic field into two components relative to a spherical coordinate system centered at the transmitter: the O-component and the r-component. The O-component is dominated by the TMi mode of propagation and the r-component corresponds to higher order TE modes as discussed by , After calculating 0, the degrees of which the NS antenna is rotated to point toward the transmitter, the rotation matrix shown in (3.1) is multiplied with the observed data to generate the true O and r magnetic field components of the transmitted signal as shown in (3.2).
cos(0) â€” sin(6)l cos(0) cos(0)
cos(0) â€”sin(0)' _cos(0) cos(0) .
3.2 Hugo, CO
Figure 3.1: Hugo, CO: Southern Site
The southern site in Hugo Colorado was 240 km south of the path of totality and 930 km from the 25.2 kHz NML transmitter in North Dakota. In Figure 3.2 both the O and r components are plotted on the same graph with the black line representing the O component and the red representing the r component. Effects can be seen on both the O and r components for the amplitude and phase during the time of the eclipse. At 17:00:54 UTC the eclipse obscuration local to Hugo was 38%. At 17:50:38 UTC a max obscuration of 92% (blue line) local to Hugo, CO was reached. At 18:57:34 UTC the eclipse was at 27% obscuration. It seems that phase exhibits change very close to the percentage of obscuration, while the amplitude seemed to behave less linearly and grew more rapidly as the obscuration increased. This could be due to a smaller â€˜path length to totality path-bandâ€™ ratio. In other words, a smaller piece of the earth ionosphere waveguide from NML to the receiver in Colorado was impacted by the solar eclipse.
Figure 3.2: Observed amplitude and phase at Hugo, CO. Vertical dotted line shows time of
The behavior of the non-dominant mode of propagation - the /'-component of the magnetic field received at Hugo, CO is more erratic. The amplitude and phase of the r-component follow the same trends as the phi-component up until totality. It is then that the amplitude of the received Br drops dramatically, to then recover again before trending downwards again like the Bo component. The inverse effect is seen in the phase plot: The Br component phase trends the same as the Bo component until the totality is reached at Lakeside where the phase begins to trend upwards rapidly until the end of totality, to then reside again until back into a normal and past-observed response to the solar eclipse. It is worth noting that a solar flare hit the earth at 17:51 UT. Possible effects of the solar flare on the observations are discussed at the end of this chapter.
3.3 Lakeside, NE
Figure 3.3: Lakeside, NE. Totality site
After seeing the data obtained at Hugo, CO and comparing it to the data obtained at
Lakeside, NE, a distinct difference in the received amplitudes and phases is seen. The Lakeside
data has more fluctuations. Also, consider the ratio of the â€” at Hugo compared to â€” at
Lakeside. The ratio of â€” a Hugo is about â€” while the ratio of â€” at Lakeside is about - . The
higher ratio at Lakeside suggest that there existed higher orders of TE modes propagating at this
location. Due to the unique position of the totality site antennas, it is believed that they were
measuring not only waves propagating away from the NML transmitter, but also waves
scattering from the eclipse induced perturbation in the ionosphere.
Observed Amplitude and Phase: Lakeside, NE
17:17:34 17:34:14 17:50:54 18:07:34 18:24:14 18:40:54 18:57:34
Figure 3.4: Observed amplitude and phase at Lakeside, NE. Vertical dotted line shows time of
maximum obscuration local to Lakeside, NE
The differences between the Hugo and Lakeside data can be attributed to the length of the transmission paths: Any interference that happened along the earth ionosphere waveguide near the totality band was diluted by the length of a less affected earth - ionosphere waveguide. Regarding the nature of the observed /'-component at Lakeside, NE, the same nature of the Br component seen in Hugo is observed at Lakeside, except for the larger fluctuations observed in the Lakeside data that its hypothesized to be due to scattering fields under the path of totality. Lastly, it is worth noting that at the times between 17:00 and 17:34 UTC in the Lakeside data you can see a dramatic step like function in the amplitude and phase. This was due to a child moving the antenna, not propagation effects.
B .(deg) EL (pT)
3.4 Solar Flare
While the solar eclipse blocked the sunlight causing electrons to recombine and thus raise the reflection height of the ionosphere, just as the sun began to show its face again to the ionosphere residing over the North American continent, a C class solar flare hit Earth.
Solar activity of Monday, 21 August 201 7
Solar flare at end of max totality (local to Lakeside, NE)
vnJ* JVaj l/ V
n lb n o n LH
21. Aug 03:00 06:00 09:00 12 00 15:00 18 00 21 00 22. Aug
Figure 3.5: Observed C class solar flare at end of max obscuration at Lakeside,
----X-ray Flux (0.1-0.8 nm)
----solar obscuration (local to Lakeside)
Figure 3.6: Both sites compared to solar flare and solar obscuration at lakeside,
Obscuration (%) Q (dgg) B^(pT) B^(deg) B(, (PT)
A concern was whether or not the solar flare was blocked by the moon so that it would not affect the ionosphere local to Lakeside and Hugo, or any part of the totality band. Based on the evidence presented in Figure 3.5 it seems that the flare could have influenced the nondominant, Br components of the VLF transmitter signal received at both Lakeside and Hugo. But, because of how the responses of the amplitude and phases to the max flare are not all exactly the same, and that the eclipse obscuration was at its maximum just before the solar flare, it is hard to conclude that the solar flare is the only contributor to the observed characteristics of the r-components. Whether or not the flare affected the r component is up for debate, but presented in chapter four is evidence of the flare causing an asymmetrical effect in the ionospheres response to the solar eclipse. One can expect that the electron density in the D region will be set by the instantaneous solar radiation. But, modeling suggests that the rate of ionization after the max totality was observed, was not the same due to the flare forcing more ionization post totality then what would have been observed during normal eclipse conditions.
4.1 Modeling the Horizontal Magnetic Field Component in LWPC
Without any post processing and modification to the LWPC code it outputs a vertically oriented electric field in dB with respect to pV/m. To find the horizontal magnetic field in pT you simply apply the following equation:
Where is go the permeability constant for free space and r| is the impedance of free space. To calculate the horizontal component using the LWPC a few more steps are required. Applying
Maxwellâ€™s equations, specifically V x E = â€” â€” in the spherical coordinate system will produce
the r-component of the B field. To do this computationally you must alter the LWPC files to compute the horizontal electric field at different altitudes so that the curl of the electric field can be calculated to then output said horizontal magnetic field component (Br). For this solution the horizontal component in question is at the surface of the earth so a curl must be computed as close to the surface as possible, thus calculations were taken at 0, 1, and 2 km to best approximate this Horizontal component:
o vertical D magnitude
â€” Ephi(0km) + 4Ephi(lkm) â€” 3Ephi(2km)
x 1012 (pT) (4.2)
where Az is the change in altitude for the three electric fields of 1000m and co is 2% x (25.2 kHz), the angular frequency of the NML transmitter. The value returned is a complex value containing the phase and amplitude.
4.2 Propagation Path Segmentation
To best capture the shape of the ionosphere it was decided to divide the Tx-Rx path into five segments as shown in Figure 4.1 (initially ten segments were chosen, it was seen that this did not perform any better than the five-segment resolution).
Figure 4.1: Boundaries
Three segments equal to 193 km connect the path from the transmitter in North Dakota to the receiver in Lakeside, NE. The path of totality was approximately 110 km wide measured from edge to edge, so the length of segment 4 is equal to said length. This allowed the use of a homogeneous profile for the totality region itself, but not for the entire eclipsed region. The step-gradients are not symmetrical in a sense of starting at zero and returning to zero because of the location of the receivers in relation to the transmitter and path of totality: the totality percentage is not only dependent on time but more importantly location. Figure 4.2 shows a discretized ionosphere at max totality due to the solar eclipse and the receiver locations.
Figure 4.2: Stepwise ionosphere segmentation compared to smoothly varying profile over the path of propagation.
4.3 h â€™ and p Step-Gradients
The next step in the process was to decide what the relative change of h â€™ and P should be for each segment. A zero step-gradient, single profile was considered first like in past work, followed by conservative h â€™ shape of | 0 | +1 |+ 2 |+ 3 | +2 | km and a p step-gradient of | 0 | +0.1 |+ 0.2 |+ 0.3 |+ 0.2 | km"1 and lastly a more aggressive h â€™ step-gradient of | 0 |+ 2 |+ 4 |+ 6 |+ 4 | km and a P step-gradient of | 0 |+ 0.2 |+ 0.4| +0.6 |+ 0.4 | km'1. Examples of these profiles are shown in Figure 4.3.
H prime Gradients per Segment
single profile â€”01232 ----02464
P Gradients per Segment
â€” single profile
Figure 4.3: Illustration of different h â€™ and p gradients per segment. Note that the Lakeside, NE receiver is at the end of boundary three (579 km from the NLM transmitter) and the receiver at Hugo, CO is at the end of boundary five (930 km
from the NLM transmitter)
4.4 Global Error Calculations
To determine which step-gradient was best, a MATLAB script was written to compute the magnetic field strength and phase at both Lakeside and Hugo for a given h â€™ and fi combination at each boundary from 68 km to 80 km and 0.38 km"1 to 0.64 km'1, respectively. The LWPC outputs were compared to observations. The differences were quantified by evaluating a global error metric. Initial modeling attempts showed that variation of the h â€™ parameter could create large changes in the received modeled amplitude. So, to not miss a possible solution in the global error map, a higher resolution was implemented in the first for-loop, providing a new solution at every 100m (0.1 km) in altitude. Because the fi parameter caused small changes to the amplitude, a more course resolution of 0.1 km"1 per iteration was selected. Next, twenty evenly space time steps from the beginning of data acquisition to the end were chosen to compare the LWPC modeled solution with. Because the data collection did not start until Lakeside was already in 35% totality, a constant shape of the ionosphere was assumed (the step-gradient). This resulted in about 400 solutions for a given time step.
To properly weight the error of each constraint at each step so that the Lakeside error did not dominate the global error calculations the analysis was done in relative phasor form, subtracting the LWPC phasor from the observed phasor divided by the magnitude of the observed phasor at each time step. Because the phase is calculated in total phase change within the LWPC, a normalization was done so that the net change could be compared.
Global error was calculated in three ways:
Complete Global Error =
, , , . , Lakeside Obs. Phase) _
> Lakeside Obs. Ampc J
) Lakeside LWPC Ampe
J^tp Lakeside LWPC Phase)
^(p Lakeside Obs. Amp
d . , â€ž pJ^r Lakeside Obs. Phase) â€” r , , . , â€ž p
3r Lakeside Obs. Amp^ Dr Lakeside LWPC Amp^
J(Br Lakeside LWPC Phase) I
r Lakeside Obs. Amp
> Hugo Obs. Ampe^* Hu9Â° 0bs' Phase > - S0//Â«5o LWPC Amp^* Hu9Â° LWPC PhaSe > I
> Hugo Obs. Amp
r Hugo Obs. Amp
Hugo Obs. Phase )_ r ,t â€ž pj^r Hugo LWPC Phase) I
? * } Â£>r Hugo LWPC Ampe a J\
r Hugo Obs. Amp
Global Error (no Lakeside Br component) =
, , , . , â€ž pj^cp Lakeside Obs. Phase) _ r , , , . , â€ž pj^cp Lakeside LWPC Phase)
y Lakeside Obs. Amp^ Â°
,t â€ž pJ^
Hugo Obs. Ampe v " ' tâ€™cfâ€™Hugo LWPC Ampe v u U
Ir ,t â€ž pJ^r Hugo Obs. Phase)_ r I7 a e>
\t>r Hugo Obs. Ampe a Hugo LWPC Ampe
j(Br Hugo LWPC Phase ) I
r Hugo Obs. Amp
Global error (no Br component) =
, , , â€ž pj(B(p Lakeside Obs. Phase)â€” p>. , . .. â€ž pj^cp Lakeside LWPC Phase)
> Lakeside Obs. Ampe ^ y a
> Hugo Obs. AmPeKB* Hu9Â° 0bsâ– Phase) - B^,Hug0 LWPC Amp^* Hu9Â° LWPC Phase > \
B(p Hugo Obs. Amp
The reason behind the three separate types of error was to see if the same solution to the two-parameter D-region electron density equation could be determined based on a global error map with all constraints or to see if leaving out the constraints where it was thought that scattering was observed was best. Figures 4.4, 4.5, and 4.6 show three global errors for the three types of profiles at time steps three, ten (max totality) and eighteen. In Figure 4.5 it can be seen
clearly that the conservative step-gradient works best and the same solution for the best fit can be determined using all three global error methods. It can be seen that the single profile and aggressive five profile model provide a similar solution and do not provide as good of a localizing solution like the conservative step-gradient five profile solution does. It was concluded from trial and error that a h â€™ step-gradient of | 0 | +1 |+ 2 |+ 3 |+ 2 | km and a fi step-gradient of | 0 |+0 .1 |+0 .2 |+0 .3 |+0 .2 km"11 provided the best fit. Note that in the following Figures the x and y axis only show 1 value of h â€™ and fi per block corresponding to the first segment. In reality, each â€˜blockâ€™ represents five segments, thus five h â€™ and /Ts. For example, if the axis shows that a good solution falls on grid point hâ€™ = 14 km andfi = 0.4 km'1, really that means the solution falls on grid pointhâ€™ = 74, 75, 76, 77, and 76 km and(1 = 0.4, 0.41, 0.42, 0.43, and 0.42 km"1
- prime (km H - prime (km) H - prime (km)
Global Error at x = 3 s
48 58 63
- 2.2 - 2 I 1.8
Figure 4.4: Three global error methods compared to different step-gradient types
at tau = 3 steps
- prime (km H - prime (km) H - prime (km)
Global Error at x = 10 s Global Error ( no Lakeside Br ) at x = 10 s Global Error (no Br ) at x = 10 s
0 (km'1) 0 (km'1) 0 (km'1)
Figure 4.5: Three global error methods compared to different step-gradient types
at tau = 10 steps
- prime (km H - prime (km) H - prime (km)
48 58 63 48 58 63 48 58 63
0 (km'1) 0 (km'1) 0 (km"1)
Figure 4.6: Three global error methods compared to different step-gradient types
at tau = 18 steps
Evidence of the solar flareâ€™s effect can be seen as the global error progress in time. As the solar eclipse was approaching max totality the solution to the two parameter ionospheric model shifted from a lower h â€™ prime and higher (J> more gradually compared to right after the totality. Once the solar eclipse began to subside and the solar flare was able to directly impact the earthâ€™s ionosphere, a much more sudden drop in h â€™ and (J> was observed, effectively pushing down the ionosphere profile faster than its resurrection during the waxing of the eclipse. This effect is illustrated in Figure 4.7. If the solar flare had not affected the ionospheric profile, the global error in time steps 7, 8, and 9 should look just like time step 13, 12, and 11, respectively. Due to the solar flare introducing more photons and disturbing the electron density, ionization was much more rapid then what it would have been in the absence of the flare.
-prime (km) H-prime (km)
Global Error at T.S 7 Global Error at T.S 8
(km'1) if (km'1)
Global Error at T.S 9
48 58 63
Global Error at T.S 11
48 58 63
Global Error at T.S 10
48 58 63
Figure 4.7: Comparison of global error for different time steps before and after
Itâ€™s easy to see that after time step 10, the ionosphere h â€™ and (J>â€™s drop faster than they rose prior to time step 10.
4.5 LWPC Solution
Once the global error script ran to completion the smallest raw error value was found each plot for the given 20 time steps, resulting in 20 sets of modeled data to compare with observed data. In Figure 4.8 the solutions within the global error plots that provided the least error in amplitude and phase are plotted.
Observed B Data vs. LWPC Model: Lakeside, NE
17:00:54 1 7:17:34 1 7:34:14 1 7:50:54
17:00:54 1 7:17:34 1 7:34:14 1 7:50:54
Figure 4.8: Observed data vs. LWPC model
The LWPC solution closely matches the amplitude observed at Lakeside but has clear discrepancies for the Br amplitude and phase components, as well as the Biphase component at Lakeside. When looking at the Br data at Lakeside, the LWPC model greatly under predicts the amplitude strength. LWPC accounts for multimodal propagation, but only in one direction. Because the ionosphere is three-dimensional other modes of propagation including possible backscatter could have constructively added together to create a stronger net Br. The increased three-dimensional scattering could be due to the solar eclipseâ€™s effect on the shape and profile of the ionosphere, as well as the solar flare that followed shortly after. These effects are also seen in the Br and B^ phase modeled components compared to the observed data at Lakeside.
The LWPC, five - segment model very accurately predicts the amplitude and phase of both the Br and B^ components when compared to the observed data at Hugo. This is the case due in part to a larger propagation distance as well as a smaller ratio of the ionosphere in the totality band compared to the ionosphere outside of the totality band. Overall, the LWPC solution was constrained by eight parameters and approximations and is not perfect, but still does a good job of modeling the dynamic nature of the ionosphere, arguably better than previous attempts even with the additional changes by the rapid day-night-day transition of the solar eclipse followed by the bombardment of a solar flare.
SUMMARY AND CONCLUSIONS
On August 21, 2017 a total solar eclipse passed over the continental United States providing a unique and rare opportunity to study the dynamic nature of the ionosphere given the rapid day-night-day transition provided by the eclipse. Observations were made at the edge of the totality band shadow on the ionosphere as well as outside of the totality band. The locations of the receivers that made the observations were positioned in a perpendicular geometry in relation to the eclipse path of totality and the position of a VLF transmitter operating under the call sign of NML. The observed data suggest that scattered fields from the totality zone may have caused an increase in amplitude in the Br component of the VLF wave. In addition to the solar eclipse, a solar flare was observed and caused a faster rate of ionization as the eclipse began to reside, compared to when it was increasing solar obscuration.
An LWPC model was created using five segments with different h â€™ and fi values at each segment to better capture the ionosphereâ€™s profile and characteristics determined by the Wait and Spies two-parameter model to calculate electron densities that ultimately affect the propagation of the transmitted VLF signal. The LWPC was unable to account for the elevated Br amplitude levels suggesting the existence of potential back scatter that constructively and destructively added to the received signal at Lakeside. The solar flare was proven to affect the ionosphere as well as providing an even more unique opportunity than previously witnessed and modeled, but now the question remains if scattering was due to a combination of the solar eclipse and flare, or
strictly due to the flare rapidly increasing the electron density and dropping the reflection height of the ionosphere faster than it climbed as the eclipse was reaching fruition.
A method of segmenting the LWPC earth - ionosphere waveguide to capture the properties of the ionosphere during a solar eclipse provided a better solution than that of a single, homogenous profile from transmitter to receiver. The implementation of a global error metric was a quick way to effectively calculate and test numerous h â€™ and P combinations so that all possible solutions for a given step-gradient selection were determined. The positions of the receivers in relation to the eclipse path and transmitter provided a new way to constraint the two-parameter model by forcing the solution to fit two separate sets of observed data due to the overlapping paths of propagation from NLM to Lakeside, and NLM to Hugo. Furthermore, scattering was observed in the data suggesting that modes of propagation are more complex than is often assumed.
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