A Brief Introduction to Ray Tracing andIonospheric Models

Stephen R. Kaeppler AD0AEskaeppl@clemson.edu

Clemson University

March 2, 2018

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Motivation - why should you care?

Prediction tool for understanding HF radiowave propagation.

Is a primary analysis tool for quantifying the ionospheric modelparameters, given a set of observations (i.e., propagationdelay, Doppler, polarization, angle of arrival (AoA).

The formulation is generalized in that it is a way to describepropagation through many media.

Important:

You are only as accurate as the ionospheric model you use in theray tracer!

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Motivation - why should you care?

Prediction tool for understanding HF radiowave propagation.

Is a primary analysis tool for quantifying the ionospheric modelparameters, given a set of observations (i.e., propagationdelay, Doppler, polarization, angle of arrival (AoA).

The formulation is generalized in that it is a way to describepropagation through many media.

Important:

You are only as accurate as the ionospheric model you use in theray tracer!

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Motivation - why should you care?

Prediction tool for understanding HF radiowave propagation.

Is a primary analysis tool for quantifying the ionospheric modelparameters, given a set of observations (i.e., propagationdelay, Doppler, polarization, angle of arrival (AoA).

The formulation is generalized in that it is a way to describepropagation through many media.

Important:

You are only as accurate as the ionospheric model you use in theray tracer!

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Motivation - why should you care?

Prediction tool for understanding HF radiowave propagation.

Is a primary analysis tool for quantifying the ionospheric modelparameters, given a set of observations (i.e., propagationdelay, Doppler, polarization, angle of arrival (AoA).

The formulation is generalized in that it is a way to describepropagation through many media.

Important:

You are only as accurate as the ionospheric model you use in theray tracer!

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Talk Organization

Where does the ray tracer come from? What are youattempting to solve?

Index of refraction: Appleton-Hartree equationSnell’s LawHamilitonian Optics formulation, aka, the Haselgrove equations

What are the di↵erent options out there for ray tracers andionospheric models?

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Appleton Hartree Index of refraction - I

The Appleton Hartree equation describes the propagation of anelectromagnetic wave in a magnetized plasma.

Starting with Maxwell’s equations:

r⇥ E =� @B

@t

r⇥ B =µ0j+ ✏0µ0@E

@t

Using a plane-wave approximation that, A / A exp(ik · x� i!t)[iis the imaginary number - physics convention], then derivativesbecome

@

@t! �i!

@

@x! ik

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Appleton Hartree Index of refraction - I

The Appleton Hartree equation describes the propagation of anelectromagnetic wave in a magnetized plasma.

Starting with Maxwell’s equations:

r⇥ E =� @B

@t

r⇥ B =µ0j+ ✏0µ0@E

@t

Using a plane-wave approximation that, A / A exp(ik · x� i!t)[iis the imaginary number - physics convention], then derivativesbecome

@

@t! �i!

@

@x! ik

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Appleton Hartree Index of refraction - II

We take the two equations from Maxwell’s equation, take the curlof Ampere’s law to arrive at the following relation:

�r⇥r⇥ E = µ0@j

@t+ ✏0µ0

@2E

@t2

which becomes,

k⇥ k⇥ ˜

E = µ0i!˜j+!2

c2˜

E

And that j = � · E and that

K = 1� �

i!✏0

we arrive at the key relation:

k⇥ (k⇥ ˜

E) + ˜

K · ˜E = 0 (1)

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Appleton Hartree Index of refraction - III

Now lets evaluate K, our goal is to arrive at a relation that is inthe form j = neeUe

, we will start with the Lorentz force for aparticle in a magnetic field, where B = B0z is in the z direction inan x,y,z coordinate system.

medUe

dt= qe(E+U⇥ B)

which in terms of components becomes and applying our relationsfor derivatives,

�i!meU1x = qe⇣E1x + U1yB0

⌘

�i!meU1y = qe⇣E1y � U1xB0

⌘

�i!meU1z = qe E1z

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Appleton Hartree Index of refraction - III

Now lets evaluate K, our goal is to arrive at a relation that is inthe form j = neeUe

, we will start with the Lorentz force for aparticle in a magnetic field, where B = B0z is in the z direction inan x,y,z coordinate system.

medUe

dt= qe(E+U⇥ B)

which in terms of components becomes and applying our relationsfor derivatives,

�i!meU1x = qe⇣E1x + U1yB0

⌘

�i!meU1y = qe⇣E1y � U1xB0

⌘

�i!meU1z = qe E1z

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Appleton Hartree Index of refraction - III

We can re-write into a matrix equation

2

4�i! �⌦e 0⌦e �i! 00 0 �i!

3

5

2

4U1x

U1y

U1z

3

5 =

2

4E1x

E1y

E1z

3

5

We can then invert this matrix and pressing forward with a fewstep to get it into j = neqe ˜

U

e

= � · ˜E,

�

i!✏0=

!2pe

i!(⌦2e � !2)

2

4�i! ⌦e 0�⌦e �i! 00 0 �i/!

3

5 (2)

where !2pe = neq2e

✏0meand ⌦e = qeB0

me

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Appleton Hartree Index of refraction - IV

Combining the Equation 1 and 2 (with a little more work), and using

n =kc

!= [n sin ✓, 0, n cos ✓]

D(k,!) · E =

2

4S � n2 cos2 ✓ �iD n2 sin ✓ cos ✓

iD S � n2 0n2 sin ✓ cos ✓ 0 P � n2 sin ✓2

3

5

2

4E1x

E1y

E1z

3

5 = 0

S = 1�!2pe

!2 � ⌦2e

D =⌦e!2

pe

!(!2 � ⌦2)

P = 1�!2pe

!2

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Appleton Hartree Index of refraction - V

Solving for det[D(k ,!)] = 0 and a few steps later (in Jackson form), weget (neglecting collisions)

n2 = 1� X (1� X )

1� X � 12Y

2 sin2 ✓ ±�(1� X )2Y 2 cos2 ✓ + ( 12Y

2 sin2 ✓)2�1/2

X =!2pe

!2=

q2ene(r , ✓,�)

me✏0!

Y =⌦e

!=

qeB0

me!

Where the O- and X-mode corresponds to the + and � in the AHequation, and corresponds to LH and RH circular polarization in thenorthern hemisphere, respectively. For ✓ = 90�, the + and � correspondsto the ordinary and extraordinary mode, respectively.

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Snell’s Law

A simple way to ray trace would be to iteratively solve Snell’s law assuminggiven that you can calculate the index of refraction, n.

n1 sin ✓1 = n2 sin ✓2

where ✓ is defined relative to the normal of the layer.

Could also derive adi↵erential form,

d(n sin ✓) = constant !dn sin ✓ � n cos ✓d✓ = 0 !

d✓dn

= tan ✓

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Snell’s Law

A simple way to ray trace would be to iteratively solve Snell’s law assuminggiven that you can calculate the index of refraction, n.

n1 sin ✓1 = n2 sin ✓2

where ✓ is defined relative to the normal of the layer. Could also derive adi↵erential form,

d(n sin ✓) = constant !dn sin ✓ � n cos ✓d✓ = 0 !

d✓dn

= tan ✓

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Snell’s Law for Spherical Symmetry

Bouger’s Rule

nb⇢b sin�b = nc⇢c sin�c

Stanford Mark X Ray Tracer.

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Lagrangian and Hamilitionian Mechanics

A well-known method for solving equations of motions of mechanical systemsare the Lagrangian and Hamilitionian mechanics formulation. The idea is basedon the concept of finding the extrema of a quantity.

�

ZL(qj , qj , t)dt = 0

“Of all the possible paths along which a dynamical system may move from onepoint to another within a specified time interval, the actual path followed isthat which is minimizes the time integral of di↵erence between the kinetic andpotential energy” [Marion and Thorton, 2004]

LagrangianFormulation

�

ZL(qj , qj , t)dt = 0

L(qj , qj , t) = T (qj)� U(qj)

dL

dqj� d

dt

dL

dqj= 0

Hamilitionian Formulation:

H(qj , pj , t) = T (qj) + U(qj)

dqjdt

=@H@pj

dpjdt

= �@H@qj

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Lagrangian and Hamilitionian Mechanics

A well-known method for solving equations of motions of mechanical systemsare the Lagrangian and Hamilitionian mechanics formulation. The idea is basedon the concept of finding the extrema of a quantity.

�

ZL(qj , qj , t)dt = 0

“Of all the possible paths along which a dynamical system may move from onepoint to another within a specified time interval, the actual path followed isthat which is minimizes the time integral of di↵erence between the kinetic andpotential energy” [Marion and Thorton, 2004]

LagrangianFormulation

�

ZL(qj , qj , t)dt = 0

L(qj , qj , t) = T (qj)� U(qj)

dL

dqj� d

dt

dL

dqj= 0

Hamilitionian Formulation:

H(qj , pj , t) = T (qj) + U(qj)

dqjdt

=@H@pj

dpjdt

= �@H@qj

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Hamiltionian Optics Formulation of Ray Tracing

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Haselgrove 1954 derived a relative general formulation describing rays beingtraced through an anisotropic medium.Starting from Fermat’s principle

�

Zµ cos↵ds = 0

Haselgrove derived the Hamilitionian:

dp

dt=

dui

dt= �g ij dH

dxi� g ijg kmul @H

@um

✓@gli@xk

� @gkl@xi

◆

dq

dt=

dxjdt

= g ij @H@ui

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Hamiltionian Optics Formulation of Ray Tracing

Algorithm was developed by Jones and Stephenson circa 1975 whichnumerically solved the Haselgrove equations. In e↵ect, the algorithmthese first order di↵erential equations in spherical coordinates:

H =1

2

✓c2

!2(k2

r + k2✓ + k2

�)� n2◆

dr

dt=

dH

dkrd✓

dt=

1

r

dH

dk✓d�

dt=

1

r sin ✓

dH

dk�

dt 0

dt=

�dH

d!

dkrdt

= �dH

dr+ k✓

d✓

dt+ k� sin ✓

d�

dtdk✓dt

=1

r

✓�dH

d✓� k✓

dr

dt+ k�r cos ✓

d�

dt

◆

dk�dt

=1

r sin ✓

✓�dH

d�� k� sin ✓

dr

dt+ k�r cos ✓

d✓

dt

◆

d!

dt=

@H

@t 0

where t is an independent parameter which will become group path(P 0 = ct 0), and k2

r + k2✓ + k2

� = !2/c2

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Hamiltionian Optics Formulation of Ray Tracing

Group Delay:

P 0 = ct 0

Phase path (time changes in phase path are Doppler shifts):

dP

dP 0 = � 1

!

kr@H@kr

+ k✓@H@k✓

+ k�@H@k�

@H@!

Can also calculate other quantities like optical path length,absorption, and polarization.

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Ionospheric Models

Important:

You are only as accurate as the ionospheric model you use in the raytracer!

Analytic Expressions: Parabolic, Quasi-parabolic, Chapman Layer(derived from theory of how ionization occurs), Bent

International Reference Ionosphere - derived from ionosonde, ISR,and many other data sources, although does use analyticexpressions and interpolation. State of the art model!

Other ”research grade” models not easily available: SAMI-3,IDA4D, etc.

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Ray Tracers, Types, and Models Used

Superdarn Raydarn: 2-D ray tracer found in Davitpy

2-D Ray tracer without magnetic field. Version of theColeman [1991 DTSO Document, Radio Science 1998]algorithm which uses a 2-D Lagrangian formulation.Advantages: Integrated with python and davitpy, parallelized,open sourceDisadvantages: only 2-D and does not include magnetic fielde↵ects

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Ray Tracers, Types, and Models Used

Proplab Pro

At least a 2-D ray tracer (probably 3-D) within IRI includingfull magnetoionic e↵ects (X and O mode).

Advantage: Most user friendly including graphics, “vetted” inamateur community.

Disadvantage: Closed source, $240(!) for the software

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Ray Tracers, Types, and Models Used

PHARLARP

CERVERA AND HARRIS: MODELING TIDs IN IONOGRAMS

Figure 1. Example of 3-D ray tracing with PHaRLAPshowing O (blue) and X (red) polarization modes and thecase for no geomagnetic field (green).

noniterative method based upon the geocentric to geode-tic latitude coordinate conversion algorithm of Mathworks[2013] which we found to have acceptable precision closeto the Earth (the solution has no error at the surface of theEarth). The error at an altitude of 400 km varies from zero atthe poles and equator to less than 13 cm at a latitude of˙45ıwhen compared to the iterative technique of Hedgley [1976].

[16] The raypath equations are a set of six coupled first-order ordinary differential equations for the ray position, Er,and the vector Ek which has a magnitude of !, the refrac-tive index, in the direction of the wave normal. They aresolved by numerically integrating them with the embed-ded fourth/fifth-order eight-step Runge-Kutta algorithm ofBogacki and Shampine [1996]. The set of Runge-Kuttaparameters (or Butcher table) developed by them yields anefficient algorithm with a high level of accuracy. Adaptivestep size control is included and is based on the error esti-mate obtained from the difference between the fourth- andfifth-order solutions. The solution returned is fourth order.

[17] When the magnetic field is ignored, the independentvariable over which the raypath equations are integrated is,conveniently, the group range, P0. However, this is not thecase for a magnetized ionosphere. The group range, for thiscase, must be calculated by integrating the group refractiveindex, !0, along the ray’s path:

P0 =Z!0 cos˛ dl (1)

where ˛ is the angle between the wave normal and ray direc-tion. This angle is easily calculated during the numericalintegration:

cos˛ =Ek ! PEr! |PEr|

(2)

The group refractive index is related to the refractive indexas follows:

!0 =d(!f )

df(3)

where f is the wave frequency of the ray. Multiplying thequadratic form of the Appleton-Hartree equation for !2 [see,e.g., Haselgrove and Haselgrove, 1960] by f 2, differentiatingwith respect to f, and after much algebraic manipulation an

equation for !0 is derived. This is then solved along with thedifferential equations for the raypath.

[18] The state vector of the ray, ES(Er, Ek), is stored at everypoint along the raypath and output once the numerical inte-gration has completed. Other quantities which are calculatedalong the raypath and output include the phase path, geo-metrical distance, refractive index, group refractive index,the angles subtended by the wave normal and ray directionand geomagnetic field, electron density, magnetic field vec-tor, magnitude of the wave polarization, and the tilt angles ofthe wave electric field vector and volume polarization vectorout of the plane of the wavefront.

[19] The various PHaRLAP algorithms are coded in For-tran compliant with the Fortran2008 specification. The rou-tines have been compiled against MathWorks mex librariesto enable them to be called within the MATLAB program-ming language for ease of use.

[20] Figure 1 shows an example of raypaths for the ordi-nary (O) and extraordinary (X) polarization modes calcu-lated using the 3-D magnetoionic ray tracing engine. Thegreen raypath in this figure is the case for no geomagneticfield. The ionosphere used in this example is sphericallysymmetric; thus, there are no ray deviations due to iono-spheric tilts. The deviation of the O-mode and X-mode raysin this case is due to the magnetic field alone.

[21] In Figure 2 we show the effect of a tilted ionosphereon the raypaths. The ionospheric tilt takes the form of anincrease in foF2 in a direction perpendicular to the launchazimuth of the rays. We immediately see that the effect ofeven a small ionospheric tilt is significant and larger thanthat due to the effect of the geomagnetic field. Figures 1and 2 clearly demonstrate that in order to model the effectsof TIDs, a full 3-D magnetoionic ray tracing procedureis required.

Figure 2. Example of 3-D ray tracing with PHaRLAPshowing the effect of a tilted ionosphere. Blue lines are afan of 5 MHz O-mode polarized rays propagated througha spherically symmetric ionosphere. The initial azimuth ofthese rays is 0ı. Note the effect of the geomagnetic fieldwhich causes a small out-of-plane deviation. The greenlines are initially the same rays as above but propagatedthrough an ionosphere which has a small ionospheric tilt( foF2 increase of 0.03 MHz per degree longitude) imposed.The red lines are the rays propagated through an iono-sphere which has a larger ionospheric tilt ( foF2 increase of0.07 MHz per degree longitude) imposed.

433

Developed by DTSO, 2-D and 3-D algorithms, including solvingHaselgrove equations with full magnetoionic e↵ects. Also includesother useful things, such as calculation of noise, etc.Advantage: Matlab interface, full bore “free” research-grade raytracerDisadvantage: Matlab interface, semi-open source (some of the keycode is precompiled), must ask for permission to use it[See Cueveraet al., 2014, Radio Science].

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Ray Tracers, Types, and Models Used

MoJo (Modified Jones-Stephenson)Radio Science 10.1002/2015RS005843

Figure 3. Three example ray fans that are used to generate the grid mesh pattern on the ground. The two purple rayfans are the north-south and east-west aligned ray fans with a beta angle of 0�. The blue fan has a beta angle of �15�.The thick lines at 0 km altitude indicate the landing position of the ray fan on the ground as a function of latitude andlongitude (in degrees).

3.2. MoJo HF Ray Trace CalculationsMoJo-15 (Modernized Jones code) is a new implementation of the original computational algorithm of Jonesand Stephenson [1975] for detailed 3-D HF ray tracing. MoJo was rewritten from the ground up in order tocapitalize on modern programming practices and computational architectures. MoJo provides a ray traceengine written in FORTRAN-90 with the main function interfaces directly callable from high-level interactivescripting languages (Python and MATLAB) for calculation automation.

For this paper MoJo-15 integrates the 3-D Haselgrove HF ray trace equations [Haselgrove and Haselgrove, 1960]in spherical 3-D coordinates, assuming the Appleton-Hartree dispersion relation with collision and the Earth’smagnetic field using a fourth-order Runge-Kutta scheme. The gradients of the electron density and magneticfield are calculated in all three directions for the Jacobian of the ray trace equations at every step of the numer-ical integration. Although MoJo-15 can also do this for the electron collision frequency, for the calculationsherein only an altitude dependent electron collision frequency model is used for simplicity.

One major feature of MoJo-15 (not included in the original Jones and Stephenson code) is the capability toquickly calculate exact modes to machine precision via the nonlinear Levenberg-Marquardt [Levenberg, 1944;Marquardt, 1963] algorithm, given a reasonable first guess of azimuth and elevation for a fixed frequency andpropagation mode (O or X). If only one mode exists connecting the source, and receiver MoJo-15 will alwaysfind this ray. In very complicated multipath environments it is a nontrivial problem to quickly identify andlocate every possible mode, particularly in time-evolving environments. The simplest brute-force approachis to integrate a very dense distribution of rays over an extended domain of azimuths and elevations andidentify any clusters of rays that land within some tolerance distance (�1 km), from the receiver. In a moreelegant approach, Cervera and Harris [2014] integrate a tessellated triangular mesh of rays and identify thosetriangles with landing points that encompass a given receiver. Once these multiple clusters are determined,a representative first guess can be provided to the MoJo homing algorithm to find the exact eigenvalues forthe mode.

Here a novel technique to elucidate how MSTIDs warp all possible propagation paths from a given transmitterto the set of all possible ground receiver locations is presented. In this technique, a series of rays is traced withinitial elevation angles ranging from �35� to +35� from zenith in the north-south plane to create a fan, whichcould also be described as a two-dimensional beam. The angles could be chosen to reflect the performanceof a particular system but, in this case, are chosen to cover the area of interest. Next, a series of Euler rotations

ZAWDIE ET AL. HF RAY TRACING THROUGH MSTIDS 1135

Developed by Kate Zawdie at NRL, basically modernized version ofJones-Stephenson algorithm.

Advantage: Full 3-D Ray Tracer, has been coupled with SAMI-3ionospheric model and other models at NRL

Disadvantage: Closed source and possibly proprietary(?)

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Ray Tracers, Types, and Models Used

Original Jones-Stephenson Algorithm

The gold standard.

Advantages: fast, modular, at least one other ray tracer basicallyuses this directly (IONORT - Italian group), source is online,GREAT Documentation!

Disadvantages: written in Fortran 4 (!), ionospheric models need tobe compiled with the code, use of common blocks, memory leaks.

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Toward an Open Ray Tracer and Summary

Hopefully have given some insight about where index ofrefraction and ray tracing equations come from and a terseoverview of the theoretical underpinning.

One strategic push forward is to develop a generalized,modernized, open source ray tracing software toolkit thatcould be both by researchers and hopefully amateurs (morework required).

Thank you!

Thank you! Questions?

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models

Toward an Open Ray Tracer and Summary

Hopefully have given some insight about where index ofrefraction and ray tracing equations come from and a terseoverview of the theoretical underpinning.

One strategic push forward is to develop a generalized,modernized, open source ray tracing software toolkit thatcould be both by researchers and hopefully amateurs (morework required).

Thank you!

Thank you! Questions?

S.R. Kaeppler A Brief Introduction to Ray Tracing and Ionospheric Models