Ionosphere Ray-Tracing of RF Signals and Solution Sensitivities to Model Parameters

Transcription

1 Ionosphere Ray-Tracing o RF Signals and Solution Sensitivities to Model Parameters Mark L. Psiaki Virginia Tech, Blacksburg, VA 2461, USA BIOGRAPHY Mark L. Psiaki is Proessor and Kevin Croton Faculty Chair o Aerospace and Ocean Engineering at Virginia Tech. He is also Proessor Emeritus o Mechanical and Aerospace Engineering at Cornell University. He holds a Ph.D. in Mechanical and Aerospace Engineering rom Princeton University. His research interests are in the areas o GNSS technology and applications, remote sensing, spacecrat attitude and orbit determination, and general estimation, iltering, and detection. ABSTRACT Methods are developed to determine the reracted propagation paths o high-requency RF signals in the ionosphere and to determine the sensitivities o these paths to changes o the input model parameters. These techniques are being developed to support the assimilation o data rom mono-static and multi-static ionosondes with the goal o improving parameterized estimates o ionosphere electron density proiles. An additional application area is that o navigation using signals rom a ground-based network o high-requency beacons. A nonlinear two-point boundary value problem solver is developed using the shooting method with Newton updates. Robust convergence is achieved by seeding the algorithm with careully designed irst guesses o the wave vector s initial aiming angles and terminal group delay. Partial derivative sensitivities o the ray-path solution are calculated using the adjoint o the two-point boundary value problem. This approach speeds the calculations when the partial derivatives o many ionosphere model parameters need to be computed. The new algorithm has been applied successully to determine the paths o O-mode and X-mode radio waves between known transmitter and receiver locations and to spitze relection points. Spitze singularities pose no diiculties or the new algorithm because it uses a Hamiltonian raypath ormulation that remains non-singular at a spitze. Copyright 216 by Mark L. Psiaki. All rights reserved. INTRODUCTION Several areas o current scientiic and engineering interest require an ability to calculate the ray paths o high-requency (HF) radio signals through an ionosphere model. They also require an ability to determine how model changes aect the ray paths. One application is the usion o ionosonde and GPS data to develop a local ionosphere model [1]. Another is the joint estimation o receiver position and ionosphere model corrections based solely on the observables o signals received rom a ground-based network o HF beacons [2]. A third application is the assimilation o data rom a network o HF beacons/receivers into a regional ionosphere model [3]. The latter two application areas are illustrated in Fig. 1. Two high-requency beacons are shown as blue dots, and a receiver is shown as a red dot. A third beacon is out o view over the horizon in eastern Canada. Three reracting ray paths are shown between the 3 beacons and the receiver, one in green, one in purple, and one in tan. Two o the ray paths undergo multiple hops, but the purple one undergoes just a single hop between the transmitter and the receiver. Reerence [2] proposes to use observables rom these ray paths to estimate the receiver position, the receiver clock oset, and corrections to a parameterized model o the ionosphere electron density proile N e ( r; p), where r is the Cartesian position vector along the ray path and p is a vector o parameters that characterize the proile. The system discussed in [3] assumes that the receiver location is known and only tries to estimation ionosphere model corrections. In this latter system, the network o HF beacons/receivers constitutes a sort o multi-static ionosonde. This paper has two main objectives. The irst one is to calculate the observables o a HF ray path that reracts through the ionosphere by solving the ray-tracing equations. These observables include the group delay, the carrier phase, and the arrival and departure directions o the signal at the two ends o the ray path. Two types o ray paths are considered. One type undergoes total relection in the ionosphere so that the transmitter and the receiver are collocated. This is the ionosonde ray- Preprint rom ION GNSS+ 216 with corrections not ound in proceedings version.

2 Fig. 1: High-Frequency beacons, a receiver, and representative ray paths that reract o o the ionosphere. path problem. The second type o ray path is a point-topoint path rom a known transmitter location to a dierent known receiver location. Ray paths o this second type can be concatenated to produce multi-hop paths. The explicit calculation o multi-hop paths is beyond the scope o this paper, except that this paper s developments or single-hop paths will provide all o the needed outputs or use in a concatenated multi-hop calculation. This paper s second main objective is to develop a means o computing the partial derivative sensitivities o the ray-path observables to problem inputs. For ionosonde relection ray paths, the needed sensitivities are the partial derivatives with respect to the elements o p, which deine the ionosphere electron density proile N e ( r; p). For point-to-point ray paths, additional calculations are developed or the partial derivative sensitivities with respect to the transmitter and receiver locations. This paper makes our contributions to the technique o RF ray-tracing. The irst is a modiied pair o Hamiltonians, dierent rom those given in [4], or use in deining the Hamiltonian dierential equations o the ray path. The second contribution is a spline method or transitioning rom the dierential equations that are based on the irst Hamiltonian to the dierential equations that are based on the second Hamiltonian. The irst Hamiltonian applies near ree space, and the second applies in regions o high electron density, i.e., near a relection point/spitze. The third contribution is a robust solver or the ray-tracing nonlinear Two-Point Boundary Value Problem (TPBVP). It uses the shooting method and Newtons method. It iteratively determines the initial wave vector direction and the inal group delay that cause the ray-path to satisy the terminal boundary conditions. This ray-path solver seeds Newton s method with good irst guesses that it generates using a simpliied ray-path model. The ourth contribution is an adjoint-based method or calculating the sensitivities o various ray-path observables to problem parameters. The calculated sensitivities are partial derivatives o the TPBVP solution observables taken with respect to input quantities that deine the TPBVP. The remainder o this paper is divided into 4 main sections plus conclusions. Section II deines the 2 new Hamiltonians, and it uses them to develop the ray-tracing dierential equations. This section also presents the two sets o boundary conditions or the two classes o ray-tracing problems that are dealt with in this paper. Section III describes the solution method or the ray-tracing nonlinear TP- BVP. It develops a combined shooting-method/newton smethod. It includes a technique or generating good irst guesses to seed the shooting/newton algorithm. Section IV derives methods or determining the partial derivative sensitivities o the ray-tracing solution and its observables with respect to problem inputs that include ionosphere parameters and the initial and inal locations o the ray path. Section V presents example ray-tracing solutions. Section VI summarizes this paper s contributions and gives its conclusions. II. HAMILTONIAN FORMULATION OF RAY- TRACING PROBLEM This section ormulates the dierential equations and the boundary conditions o two ray-tracing problems. One is an ionosonde relection problem, and the other is a pointto-point problem. 2

3 A. Hamiltonian Dierential Equations A ray-tracing solution can be characterized by the time histories o its Cartesian position vector r and its Cartesian wave vector k. They obey the Hamiltonian dierential equations [4]: ( ) T H d r dp = k H k ( ) T H d k dp = r (1) H k where H is an appropriate Hamiltonian, P = ct group is the range-equivalent group delay, and k = ω/c is the ree-space wave number with ω being the transmission requency. The transpose operations in these equations compensate or the act that the two gradients o the scalar Hamiltonian H with respect to the column vectors r and k are deined to be row vectors. The same Hamiltonian can be used to develop a dierential equation or the range-equivalent carrier phase P = φ/k with φ being the carrier phase in radians. This dierential equation takes the orm dp dp = ) k k (2) H k k ( H The pair o dierential equations in Eq. (1) can be used to develop the ollowing state-space orm o the ray-tracing dierential equations where the state vector is dx dτ = P dx dp = (x, P, p) (3) x = [ r/p ] k/k and where P is the total group delay rom the initial time to the inal time along the ray path. The independent variable τ = P /P is the ractional group delay along the ray path. It is non-dimensional and ranges rom τ = at the start o the ray path to τ = 1 at the end. The 6-element state vector x is non-dimensional, as is the 6- element vector unction (x, P, p). This unction can be derived rom the right-hand sides in Eq. (1). (4) The range-equivalent carrier phase can be appended to the state vector in order to develop a 7-element state as ollows: r/p x = k/k (5) P/P I using this latter state deinition, then the (x, P, p) unction in the state dierential equation must be augmented to become a 7-element unction by including a inal term that is based on the scalar carrier-phase dierential equation in Eq. (2). This augmented 7-element state is not needed in order to solve or a ray path, but it is needed i one wants to compute the carrier-phase observable. B. Two New Appleton-Hartree Hamiltonians The ray-tracing dierential equations use Hamiltonians that are derived rom Maxwell s equations or radio wave propagation in a lossy magnetoplasma. They assume that the wavelength is vanishingly small relative to the spatial scales o the variations o the electron density, the Earth s magnetic ield, and the electron-neutral collision requency. This is the same analysis that has been used to derive the Appleton-Hartree index o reraction [5]. The ray-tracing solution ensures that k /k equals the Appleton-Hartree index o reraction in the lossless case. Reerence [4] presents several Hamiltonians that are based on the Appleton-Hartree index o reraction or on the underlying analysis that produces it. The ollowing new Hamiltonian is closely related to the one deined in Eq. (21) o [4]: H 1 ( r, k, k, p) = [( 1 k T ) ] 2 real{ k k n 2 AH( r, k, k, p) k 2 [(1 jz) 2 Y 2 ](1 jz)} (6) where n AH is the lossy Appleton-Hartree index o reraction [5], Y = ω eg /ω is the ratio o the unsigned electron gyro requency to the transmission requency, and Z = ν/ω is the ratio o the electron-neutral collision requency to the transmission requency. The inal two actors on the right-hand side o this equation, the terms that involve Y and jz, make this Hamiltonian dierent rom that in Eq. (21) o [4]. They have been introduced in order to make this Hamiltonian similar the second one that is used in this paper s developments. Similar to its counterpart in [4], the Hamiltonian in Eq. (6) is used near ree space, where the electron density N e ( r; p) is relatively small. It is not appropriate near an 3

4 ionosonde relection point/spitze singularity. This Hamiltonian has singularities at such points. Thereore a second Hamiltonian is used near such points, exactly as in [4]. This paper s new Hamiltonian or use near a spitze is signiicantly dierent rom the corresponding Hamiltonian given in Eq. (22) o [4]. The new Hamiltonian equals one hal the real part o an eigenvalue o a 3 3 matrix. This matrix constitutes the coeicient that multiplies the electric ield vector in a homogeneous equation that is derived using Maxwell s equations. For a given direction o the wave vector, the square o the Appleton-Hartree index o reraction can be determined by solving a generalized eigenvalue problem or the matrix. This matrix is D( r, k, k, p) = [( k T ) k 1 I k k T ] k 2 k 2 [(1 jz) 2 Y 2 ](1 jz) + X{(1 jz) 2 I Y 2 ˆB ˆBT j(1 jz)y [ ˆB ]} (7) where X = ω 2 p/ω 2 is the square o the ratio o the plasma requency to the transmission requency and where ˆB = B / B is the unit vector that points in the direction o the ambient geomagnetic ield vector B. The notation [ ˆB ] indicates the 3 3 skew-symmetric cross-productequivalent matrix. It produces the same result when multiplying a vector on the let as would be produced by taking the cross product o that vector with ˆB on the let. The second Hamiltonian takes the orm: H 2 ( r, k, k, p) = 1 real(λ) (8) 2 where λ is an eigenvalue o the D matrix: [ D( r, k, k, p) λi] E = (9) and E is the associated electric ield eigenvector This Hamiltonian does not experience any singularities at a relection point/spitze. Thereore, it is useul when the electron density is high. This Hamiltonian is not useul when the electron density is small because o diiculties in distinguishing the correct eigenvalue/eigenvector pair to use o the 3 that exist. The two Hamiltonians have dierent methods o enorcing selectivity or the O mode or the X mode o radio wave propagation. For H 1, this selection is enorced by the choice o a sign in the Appleton-Hartree ormula or n 2 AH [5]. For H 2, this selection is enorced via proper choice o the eigenvalue/eigenvector pair. In the present context, this choice is made by enorcing continuity o the eigenvector direction E along the ray path. Abrupt changes are ruled out. The initial selection o an appropriate direction at the change rom using H 1 to using H 2 is accomplished by computing the same eigenvector rom its corresponding Appleton-Hartree ormula [5] whenever H 1 is being used. When enorcing continuity, one must be careul to compute a sensible directional dierence between two neighboring complex-valued E vectors along a given ray path. The new H 2 Hamiltonian is conjectured to be more reliable at maintaining the proper mode selection than the one in Eq. (22) o [4]. The latter Hamiltonian has no explicit mode selection mechanism. It relies on dierential equation solution continuity to select the correct wave propagation mode. The new H 2 Hamiltonian relies on a modiied version o continuity, that o its associated electric ield eigenvector. It is conjectured that this orm o continuity provides stronger guarantees o proper mode selection. The dependence o H 2 on the position vector r and the ionosphere parameter vector p comes partly through the dependencies o X, Y, and Z on these quantities. These dependencies take the orms: X = N e( r; p)q 2 e m e ε ω 2 Y = B ( r) q e m e ω Z = ν( r) ω (1) where q e is the electron charge, m e is the electron mass, ε is the permittivity o ree space, and ν( r) is the electronneutral collision requency. Additional H 2 dependence on r comes through the dependence o ˆB on r. The Earth magnetic ield model B ( r) used in the present study is the IGRF ield [6]. One must calculate partial derivatives o an eigenvalue o the D( r, k, k, p) matrix in order to compute the partial derivatives o H 2 that are needed in Eqs. (1) and (2). The ollowing linear system o equations can be solved in order to compute the partial derivatives o the eigenvalue λ and the associated eigenvector E with respect to the quantity η: [ ] (D λi) E E T E η λ η = D E η (11) The notation () T indicates the complex conjugate o the transpose o the vector in question. The scalar dierentiation variable η is any quantity on which D depends, 4

5 i.e., any component o r, any component o k, k, or any component o p. The second line in this equation ensures that the (possibly complex-valued) E vector does not change its norm and that it does not undergo any unnecessary simultaneous complex phase rotations o all o its elements. This eigenvalue dierentiation technique breaks down i λ is a repeated eigenvalue o the D matrix. Repeated or nearly repeated eigenvalues occur near ree space, when X is small. That is why Hamiltonian H 2 is used only in regions o high X, which correspond to regions o high electron density N e ( r; p). C. Transition Between the Two Hamiltonians The Hamiltonians H 1 and H 2 yield identical ray-tracing dierential equations in the lossless Z = case. The trailing actors that involve Y and jz in the H 1 Hamiltonian o Eq. (6) have been included in hopes o making the two Hamiltonians dierential equations more similar in the lossy case o nonzero Z. The ollowing rationale inspired the inclusion o these terms: In eect, these same actors multiply the D( r, k, k, p) matrix o Eq. (7). The net eect is that the D matrix eigenvalues are multiplied by these same actors, which makes them similar to the ormula or H 1 in Eq. (6). Nevertheless, these two Hamiltonians will yield slightly dierent (x, P, p) unctions in Eq. (3). Let them be designated as 1 (x, P, p) and 2(x, P, p). It is necessary to transition rom 1 to 2 as X increases along a ray path. The transition method used in [4] is to switch abruptly at X =.1. The transition used here is dierent. It takes the gradual orm: 1 (x, P (x, P, p) i X X 1, p) = T (x, P, p, X) i X 1 < X < X 2 2 (x, P, p) i X 2 X (12) where T (x, P, p, X) = 1 (x, P, p) + s(x)[ 2 (x, P, p) 1 (x, P, p)] (13) with ( ) 2 [ ( )] X X1 X X1 s(x) = 3 2 X 2 X 1 X 2 X 1 (14) Equations (12)-(14) implement a cubic spline transition rom using 1 (x, P, p) below X 1 to using 2 (x, P, p) above X 2. The cubic spline unction S(X) implements this transition. It changes smoothly rom to 1 over the interval rom X 1 to X 2, and it has zero-valued derivatives at X 1 and X 2. The resulting composite (x, P, p) unction in Eq. (12) is continuous with continuous irst partial derivatives with respect to x i each o its constituent unctions 1 (x, P, p) and 2(x, P, p) are similarly smooth. The X transition range limits X 1 and X 2 are user selectable. They must obey X 1 < X 2. For an O-mode ray path they must be less than 1. Typical values might be X 1 =.4 and X 2 =.6. For an X-mode ray path, the allowable values o X 1 and X 2 depend on the magnitude o Y. Consider the case Y > 1, i.e., the case where the transmission requency is lower than the electron gyro requency. X 1 and X 2 both must be less than 1 in this situation, and the values X 1 =.4 and X 2 =.6 are reasonable. For Y < 1, the transition range limits must be less than the relection value X rel = 1 - Y. Typical values used in this study are X 1 =.4(1 Y ) and X 2 =.6(1 Y ) when calculating an X-mode ray path with Y < 1. D. Ray-Tracing Boundary Conditions Each o the two ray-tracing problems has initial and terminal boundary conditions. There are 4 scalar initial boundary conditions. They apply at τ = and take the orm = [I, ]x() r P = H 1 ( r, k(), k, p) = H 1 ( r, {[, k I]x()}, k, p) (15) The irst line o this equation sets the transmitter position at r. The second line ensures that the initial wave vector has magnitude equal to the Appleton-Hartree index o reraction multiplied by k. Ater initialization at, the Hamiltonian will remain zero due to the orm o the Hamiltonian dierential equations in Eq. (1). There are 3 scalar termination boundary conditions at τ = 1. For the ionosonde relection case, these conditions take the orm: = k(1)/k = [, I]x(1) (16) The terminal boundary conditions or the point-to-point case take the orm = [I, ]x(1) r P (17) where r is the known receiver location at the end o the ray path. 5

6 The terminal boundary conditions or both cases can be written in the general orm: = A x(1) + b P (18) where A is a 3 6 matrix and b is a 3 1 vector. This orm o the terminal boundary conditions will be used to develop the ray-tracing solution algorithm in Section III. The initial and terminal boundary conditions can be rewritten in the ollowing general orms: = g [x(), P, p, r ] = g [x(1), P, r ] (19) with the understanding that g will be independent o r or the ionosonde relection case o Eq. (16). These orms o the boundary conditions will be used to develop the ray-tracing sensitivity calculations in Section IV. III. RAY-TRACING SOLUTION METHOD The ray-tracing problem is a nonlinear TPBVP. There is no closed-orm solution or the general case. Thereore, numerical methods have been developed to solve this problem. The main solution algorithm combines the shooting method to deal with the terminal boundary conditions and Newton s method to deal with the nonlinearities. The shooting method works but guessing the ree initial conditions, numerically integrating orward to the terminal condition, checking whether the terminal conditions are solved, incrementing the guess o the ree initial condition components in a way that will reduce the error in the terminal boundary conditions, and iterating this process until the ree initial state vector components converge to values that cause the terminal boundary conditions to be satisied. Newton s method provides a means o computing the needed increments to the ree components o the initial state vector. It converges very rapidly when it gets near the solution. Newton s method can ail to converge i its guess starts too ar rom the solution. Two strategies are employed to avoid this mode o solution ailure. The irst strategy is to develop a reasonably good irst guess o the ree initial state vector components. The second strategy is to employ the Levenberg-Marquardt technique [7] as a means or orcing the Newton algorithm to converge to a local minimum o the sum o the squares o the errors in the terminal boundary conditions. 6 A. Newton s Method Implementation o the Shooting Algorithm The Newton s method implementation o the shooting algorithm or the TPBVP solves 3 equations in 3 unknowns. The 3 unknowns consist o the 2 ree parameters o the unknown direction o the initial wave vector ŝ = k()/ k() and the unknown range-equivalent terminal group delay P. The ollowing algorithm implements a Newton/shooting method or reining initial guesses o ŝ and P until the terminal boundary conditions are satisied: 1) Obtain initial guesses o ŝ and P. 2) Solve or the value o the initial wave vector magnitude k init that satisies the initial condition on the Hamiltonian = H 1 ( r, k init ŝ, k, p). This equation is quadratic in k init and can be solved in closed orm. 3) Set the initial ray-tracing state vector to [ ] r /P x() = (2) (k init /k )ŝ 4) Numerically integrate the 6-element orm o the raytracing state dierential equation in Eq. (3) rom τ = to τ = 1. Use a Runge-Kutta method such as the Dormand-Prince 4/5 method [8]. Use predeined step sizes here. Adaptation will take place elsewhere. 5) Check whether the norm o the terminal boundary condition A x(1) + b /P is suiciently small. I so, then stop. Otherwise, proceed to the next step. 6) Compute two allowable perturbation directions or (k init /k )ŝ by determining ˆv 1 and ˆv 2 such that [ˆv 1, ˆv 2, ˆv 3 ] is an orthonormal triad with ( ) T H 1 k ( r,k initŝ,k,p) ˆv 3 = H 1 (21) k ( r,k initŝ,k,p) ˆv 1 and ˆv 2 can be determined by computing an orthonormal/upper-triangular actorization o ˆv 3 [7]. 7) Initialize the 6 3 linearized solution sensitivity matrix [ ] r /(P Γ () = )2 (22) ˆv 1 ˆv 2 8) Numerically integrate the ollowing linear sensitivity matrix ordinary dierential equation rom τ = to τ = 1 dγ (τ) dτ = F (τ)γ (τ) + [,, u(τ)] (23)

7 where F (τ) = x u(τ) = [x(τ),p,p] P [x(τ),p,p] (24) This matrix dierential equation must be numerically integrated using the same Runge-Kutta method as is used in Step 4. It must use identical steps, and it must be integrated in a manner that is identical to what would occur had it been integrated simultaneously with x(τ) in Step 4 as though x and the 3 columns o Γ ormed one large state vector with 24 elements. 9) Solve the ollowing linearized terminal boundary condition or the initial wave vector increment parameters α 1 and α 2 and or the terminal group delay increment P : where α 1 = A bc α 2 + b bc (25) P A bc = {A Γ (1) + [,, b bc = A x(1) + b P b (P ]} )2 (26) 1) Determine a new initial direction o the wave vector and a new terminal group delay: ŝ new = (k init/k )ŝ + ˆv 1 α 1 + ˆv 2 α 2 (k init /k )ŝ + ˆv 1 α 1 + ˆv 2 α 2 P new = P + P (27) 11) Replace ŝ by ŝ new and P by P new and return to Step 2 The irst partial derivative o with respect to x in Eq. (24) requires second partial dierentiation with respect to r o the problem model unctions N e ( r; p), B ( r), and ν( r). This is true because the computation o requires irst partial dierentiation with respect to r as per Eq. (1). Thereore, these second derivatives must exist and be continuous. One way to achieve the needed smoothness o N e ( r; p) is to employ a model that works with smooth vertical proiles. The parameters o the vertical proiles are modeled as 2 nd -order smooth unctions o latitude and longitude through the use o a bi-quintic latitude/longitude spline. An example N e ( r; p) proile o this type is discussed in [9]. B. Levenberg-Marquardt Modiications to Newton s Method I the current guesses o ŝ and P are ar rom the solution, then the terminal boundary condition error norm computed in Step 5 o the Newton algorithm may increase or remain the same relative to the previous step. It should decrease in order to ensure convergence to a solution. A decrease in the Step-5 norm can be ensured through use o the Levenberg-Marquardt method or computing the solution increments [7]. The Levenberg-Marquardt method determines α 1, α 2, and P by a means that diers rom the linear equation solution o the Newton algorithm s Step 9. They are determined using the ormula: α 1 α 2 = (A T bca bc + γi) 1 A T bcb bc (28) P or some non-negative value o the scalar Levenberg- Marquardt parameter γ. It is easy to veriy that the Levenberg-Marquardt values or α 1, α 2, and P equal the Newton s-method values rom Step 9 when γ =. For larger γ values the increments tend to decrease, and they become vanishingly small as γ. It is not diicult to prove that there must exist a inite nonnegative value γ such that the computed increments rom Eq. (28) cause the terminal boundary condition error norm to decrease in Step 5 o the algorithm. The Levenberg-Marquardt method can be incorporated into the Newton algorithm o the previous subsection as ollows: Each major outer iteration loop rom Steps 2 through 11 is augmented by an inner iteration loop that implements a tentative set o calculations rom the Step-9 increment calculation to the subsequent Step-5 test o the terminal boundary condition error norm. I the error norm o Step 5 has not decreased relative to its value on the last major iteration, then the operations are repeated starting rom the preceding Step 9, except that the Levenberg- Marquardt increment calculation in Eq. (28) is used in place o the solution o the linear system in Eq. (25), and this calculation uses an increased γ value. The value γ = is tried or the initial increment calculation o each major outer iteration in order to attempt a ull Newton step. The ull Newton step yields ast convergence near the solution, which is why it is preerred. I the initial Newton step ails to decrease the terminal boundary condition error, then an alternate set o α 1, α 2, and P increments are computed using a positive value o γ. This initial positive value is chosen to equal a small raction o the square o the smallest singular value o the matrix A bc or to equal an even smaller raction o the square o the largest singular value o this matrix, 7

8 whichever is larger. I the resulting increments α 1, α 2, and P still ail to reduce the terminal boundary condition error norm at Step 5, then the current positive value o γ is increased through multiplication by a actor o 3. The steps rom 9 through the subsequent Step 5 are again repeated. This inner iteration repeats itsel until the terminal boundary condition error norm is decreased. At this point, the inal successul value o γ may be larger than the irst non-zero trial value by several actors o 3. Finally, the values o ŝ new and P new replace the original ŝ and P values. Prior to achieving a decrease o the terminal boundary condition error, the proposed ŝ new and P new are only accepted as tentative replacements or the original ŝ and P rom the previous major outer iteration o the algorithm. C. Generation o a Ray-Tracing First Guess It is important to supply a reasonably good irst guess to the Newton s method implementation o the shooting algorithm. Even with the Levenberg-Marquardt modiication, a good irst guess is important. I the irst guess is ar rom the solution, then the algorithm may require many iterations to converge. It may even converge to a nonzero local minimum o the terminal condition error norm, which is not a TPBVP solution. The initial guess algorithm starts with knowledge o the ray path initial position r and the ray path terminal position r. For the point-to-point case r is a given o the problem. For the ionosonde relection case r is set equal to r. The initial guess is generated in 3 stages. The irst stage produces a coarse estimate o a peak relection point along the ray path. Call it r r. Three conditions serve to determine this point. Beore deining these conditions, it is helpul to deine three unit direction vectors that all depend on r r : ŝ = r r r r r r ŝ = r r r r r r ( T h alt ŝ ra = r (29) rr) where h alt ( r) is the altitude o the Cartesian position vector r above the WGS-84 Earth ellipsoid [1]. The unit direction vector ŝ points along the initial ascending straight-line ray path approximation, ŝ points along the inal descending straight-line ray path approximation, and ŝ ra points in the local vertical direction at the approximate relection point r r. These three vectors can be used to develop the ollowing system o 4 scalar equations in 4 unknowns: = ŝ ra η 2 (ŝ ŝ ) = n 2 AH( r, ŝ )[1 (ŝ T raŝ) 2 ] n 2 AH( r r, ŝ ) (3) The irst o these equations is a 3 1 vector equation, and the last equation is a scalar equation. The our unknowns o this system are the three components o the relection position r r and the scalar parameter η. These equations model total internal relection at an abrupt change in reractive index. The vector equation assumes that the relective surace is locally horizontal. It enorces the relection conditions that ŝ, ŝ, and ŝ ra be coplanar and that the angle o incidence equal the angle o relection. The scalar equation amounts to Snell s law or total internal relection when the index o reraction changes abruptly rom its value at r to its value at r r. All o this change is assumed to take place at r r. The Appleton- Hartree index o reraction ormula used in Eq. (3) is the lossless version (i.e., Z = ); it returns real values o n 2 AH. Solution o this system o equations produces an initial coarse guess o r r. The system o equations in Eq. (3) is nonlinear and must be solved iteratively. This nonlinear system o equations is also solved using Newton s method, but special precautions are taken to ensure that the solution converges robustly. Initially a dierent set o equations is solved. It includes the irst equation, but it replaces the scaler 2 nd equation by the equation h r = h alt ( r r ) or a pre-speciied value o h r. This amounts to solving or a proposed ray path with the known maximum altitude h r rather than a ray path that satisies Snell s law or total internal relection. It is airly easy to solve this alternate set o equations by using a spherical approximation o the Earth to generate a irst guess and aterwards implementing a Newton s method solution starting rom that guess. The result o this solution procedure is the candidate relection point r r (h r ). Next, a grid o candidate h r values is deined, and a set o candidate relection points r r (h r ) is generated or this grid. The right-hand side o the second equation in Eq. (3) is evaluated at these grid points, and the algorithm looks or the lowest-altitude grid points where the error in this latter equation changes sign. These two points are then used to seed a 1-dimensional guarded Newton s-method search along h r or the point where the second equation in Eq. (3) is accurately satisied. The second stage o the ray path irst-guess calculation solves an alternate system o equations. It is similar to the system in Eq. (3), except that it replaces the two 8

9 instances o ŝ ra with a dierent relection surace unit normal vector: ( (n 2 AH ) ) T r ( rr,ŝ,k,p) ŝ rb = (n2 AH ) (31) r ( rr,ŝ,k,p) This is the unit normal to the surace o constant index o reraction that passes through the relection point r. This alternate system o equations discards the assumption that the abrupt change in index o reraction occurs normal to a constant-altitude surace. It makes the reasonable assumption that the change occurs normal to a surace o constant index o reraction. The resulting system o 4 equations in the 4-element vector o unknowns [ r r ; η] is solved using Newton s method. The Newton iteration is seeded with the candidate relection point that is the solution o the previous system in Eq. (3). This Newton iteration tends to converge rapidly because the irst guess is very good or typical ionosphere models where the predominant electron density gradients occur along the vertical axis. This second r r solution could be used to generate the required initial guesses or the Newton s method shooting algorithm o Subsection III-A. They take the orm: ŝ = ŝ { P rr r = ionosonde case r r r + r r r point-to-point case (32) The third stage o the irst-guess calculations starts by generating 5 candidate initial wave vector directions. They are ŝ a = ŝ ŝ b = cos(θ)ŝ + sin(θ)ˆv 1 ŝ c = cos(θ)ŝ sin(θ)ˆv 1 ŝ d = cos(θ)ŝ + sin(θ)ˆv 2 ŝ e = cos(θ)ŝ sin(θ)ˆv 2 (33) where the unit direction vectors ˆv 1 and ˆv 2 are chosen so that [ˆv 1, ˆv 2, ŝ ] is an orthonormal triad. The aiming oset angle θ is set to 1 deg. These 5 candidate directions include and bracket the candidate initial wave vector direction ŝ that has been generated in the previous stage o the initial guess calculations. The inal stage o the initial guess calculations numerically integrates the 6-state version o the ray-tracing system in Eq. (3). It uses the guessed value o P rom Eq. (32) in this integration, but it integrates twice as ar as the nominal terminal time: It integrates rom τ = out to τ = 2. It then perorms a brute-orce search or the τ value that solves the ollowing optimization problem: ind: τ to minimize: J(τ) =.5 A x(τ) + b /P 2 subject to: 2/3 τ 2 (34) This optimization is used to reine the initial guess o P by inding the point along the guessed ray path that comes closest to satisying the terminal boundary conditions. I τ opt is the optimal solution to the problem in Eq. (34), then P new = P τ opt (35) is a better initial guess o P. This calculation o ray paths and reinement o P is carried out 5 times using the 5 dierent guesses o the initial wave vector direction deined in Eq. (33). Suppose that these 5 ray paths produce 5 dierent optimized values o the cost unction in Eq. (34). Let them be designated as J opta, J optb, J optc, J optd, and J opte. The chosen guess o ŝ will be the candidate rom Eq. (33) that produces the lowest optimal terminal boundary condition error cost, and the chosen guess o P will be the corresponding P new value rom Eq. (35). D. Numerical Integration Step Size Adaptation It is advisable to implement step size adaptation when using a Runge-Kutta numerical integration algorithm. The Dormand-Prince 4/5 algorithm has been used in this study [8]. It is the same algorithm as is used in the popular MAT- LAB unction ode45.m. It supports step size adaptation. When using a numerical integration algorithm within an outer numerical solver or a TPBVP, however, step size adaptation can be harmul. It can cause a violation o the assumptions that go into the Newton s method linearization o the terminal boundary conditions in Eq. (25). Such violations can cause problems when trying to orce the errors in the terminal boundary conditions to decrease or each major iteration o the algorithm o Subsections III-A and III-B. The methods developed in this paper employ Dormand- Prince step size adaptation in two places. The irst place is during the numerical integration calculations that are associated with the third stage o the initial guess generation procedure. These are the 5 integrations that start rom the 5 candidate initial wave vector directions o Eq. (33). The sequence o Dormand-Prince Runge-Kutta step intervals associated with the best initial wave vector direction is then chosen as the ixed set o intervals or use in the Newton/shooting TPBVP solver. Suppose that the numerical integration grid points which have been used to 9

10 generate the best irst guess are τ, τ 1, τ 2,..., τ k,..., τ N. Note that τ = and τ N = 2 by design o the trial integration and that τ 1,..., τ N 1 are chosen by the step size adaptation algorithm. Suppose, also, that τ k = τ opt, the solution o the optimization problem in Eq. (34). Then the change rom P to P new in Eq. (35) necessitates the ollowing change in the Dormand-Prince numerical integration grid points: τ inew = τ i /τ k or all i = 1,..., k (36) These are the numerical integration grid points that are used in the TPBVP solution. It can be helpul to use two sets o numerical integration grid points in order to speed algorithm execution while achieving a speciied level o numerical integration accuracy in the TPBVP solution. The idea is to do an initial TPBVP solution which uses a coarser set o numerical integration grid points that does not achieve the inal desired level o integration accuracy. Aterwards, a second TPBVP solution is computed using a revised sequence o numerical integration grid points that will deliver the desired accuracy. In this scenario, a coarse numerical integration error tolerance is used to generate the initial solution guesses described in the previous subsection along with the corresponding integration grid points deined in Eq. (36). These grid points are then used to solve the TPBVP. Aterwards, the TPBVP solution values o ŝ and P are input to an adaptive step size Dormand-Prince numerical integration rom τ = to τ = 1. This second adaptive numerical integration uses the inal target error tolerance to control its adaptation. The resulting numerical integration grid points are used as a ixed set o grid points in a second solution o the TPBVP. The irst guesses o ŝ and P or this second TPBVP solution are the inal solution values rom the problem that used the coarser numerical integration grid. Experience with this approach indicates that the second call o the TPBVP solver converges to a very accurate solution in a single Newton iteration. IV. SENSITIVITIES OF THE RAY-TRACING SOLUTION AND ITS OBSERVABLES Sometimes one needs to compute the partial derivatives o TPBVP solution outputs with respect to TPBVP inputs. The local ionosphere data assimilation problem o [1] and the regional ionosphere data assimilation problem o [3] could useully employ the partial derivatives o group delay and carrier phase with respect to ionosphere parameters, i.e., P / p and P (1)/ p. A joint navigation/ionosphere-correction algorithm like that proposed in [2] could use P / p and the group delay partial derivatives with respect to the receiver location P / r. An algorithm that concatenates multiple raypath hops needs the partial derivatives o the initial and inal wave vectors with respect to the initial and inal raypath positions k()/ r, k(1)/ r, k()/ r, and k(1)/ r. This section develops methods or calculating these partial derivatives. A. Direct Calculation o Partial Derivatives with Respect to Ionosphere Parameters The direct method o calculating any given partial derivative starts by taking the partial derivative o the entire TPBVP. This includes dierentiation o the ordinary dierential equation in Eq. (3) and dierentiation o the initial and terminal boundary conditions in Eq. (19). Suppose that one wants to calculate partial derivatives with respect to the i th element o the ionosphere parameter vector p. Then the direct method results in the ollowing linear TPBVP or the unknowns P / p i and x(τ)/ p i on the interval τ 1: [ ] d x(τ) = F (τ) x(τ) + u(τ) P + u i (τ) dτ p i p i p i x() P = G + w + w i p i p i where = G x(1) p i u i (τ) = p i G = g x w = g w i = g p i G = g x + w P p i (37) [x(τ),p,p] [x(),p,p, r] P [x(),p,p, r] [x(),p,p, r] w = g P [x(1),p, r ] [x(1),p, r ] (38) The linear TPBVP in Eq. (37) can be solved using state transition matrix methods that are airly standard. Such methods can be used to develop a ormula or x(1)/ p i that involves a non-homogeneous term and terms that are linear in x()/ p i and P / p i. This expression 1

11 can be substituted into the third line o Eq. (37). The resulting equation, when combined with the second line o Eq. (37), yields a well-deined linear system o equations or the unknown quantities x()/ p i and P / p i. The solution or these quantities and the ODE in the irst line o Eq. (37) can be used along with numerical integration to determine x(τ)/ p i at another other value o τ. B. Rationale or Using Adjoint Calculation o Partial Derivatives The calculations o the preceding subsection are straightorward to implement. Note, however, that a new linear TPBVP would have to be solved or each element o the parameter vector p. The only change rom element to element would be the non-homogeneous terms in the TPBVP. Nevertheless, the computational cost o re-solving many TPBVPs could be prohibitive. An alternate approach to partial derivative calculation uses an adjoint. Beore developing such an approach or the TPBVP in Eq. (37), it is useul to review the reason why an adjoint calculation can reduce the number o computations. Consider a simple vector space example o using an adjoint. Suppose that the n x 1 vector x is an implicit unction o the n p 1 vector p through the n x 1 vector unction : = (x, p) (39) and suppose that the n y 1 vector y is an explicit unction o x: y = h(x) (4) The partial derivative o y with respect to p can be calculated by irst taking the partial derivative o Eq. (39) with respect to p and using the result to solve or x/ p. Aterwards, the partial derivative o Eq. (4) can be taken with respect to p, and the ormula or x/ p can be substituted into the result in order to yield: ( [ ] ) 1 y p = h x = ( h x x p [ ] ) 1 x p (41) The only dierence between the right-hand sides on the two lines o this equation is the order o their matrixmatrix multiplications. The upper line is the direct-method calculation o y/ p, and the lower line is the adjoint calculation. The transpose o the term in parentheses on the second line is the calculation s adjoint. These two versions o the same matrix-matrix multiplications have dierent operation counts. The direct method on the irst line employs n 2 xn p + n y n x n p scalar multiplications. The adjoint method on the second line employs n 2 xn y +n y n x n p multiplications. I n y < n p, then the adjoint method is more eicient. I n y n p and n x is large, then the computational burden o the adjoint method may be orders o magnitude lower than that o the direct method. Similar considerations carry over to the sensitivities o the solution o this paper s TPBVP. The corresponding potential or computational cost savings constitutes the rationale or developing an adjoint method to compute the TPBVP s partial derivatives. C. Output Functions to be Dierentiated The preceding discussion justiies the use o an adjoint method i the number o problem outputs is much lower than the number o inputs. Furthermore, the adjoint itsel is calculated using partial derivatives o the outputs. Thereore, it is necessary to deine the TPBVP outputs whose partial derivatives are to be computed. The problem outputs o interest are the range-equivalent total group delay P, the range-equivalent terminal carrier phase P (1), the initial wave vector k(), and the terminal wave vector k(1). Output o the terminal carrier phase implies that the adjoint calculations must work with a state-space model or the 7-element state vector that is deined in Eq. (5). Thereore, all developments in this section presume the use o this state vector and use o the corresponding 7-element (x, P, p) dynamics model unction. These developments also assume that the raytracing TPBVP solution values or x(τ) and P have been computed. Use o the 7-element state vector necessitates a modiication o the generalized initial condition unction g [x(), P, p, r ] in the irst line o Eq. (19). It must be augmented by appending the expression [,,,,,, 1]x(), which changes it rom a 4-element vector unction into a 5-element unction. This added initial condition speciies that the initial carrier phase is so that P (1) gives the range-equivalent change in the carrier phase over the ray path. Given the 7-element orm o x(τ), all desired outputs o the TPBVP can be written in terms o the initial state x(), the terminal state x(1), and the terminal range-equivalent group delay P. The ollowing two general ormulas cover all possible desired outputs: y = h [x(), P ] y = h [x(1), P ] (42) 11

12 Adjoint partial derivative calculations will be developed or these two generalized output vectors. In preparation or developing the adjoint method, it is helpul to deine the ollowing partial derivatives o the vector unctions h and h : H = h x [x(),p ] z = h H = h x P [x(),p ] z = h [x(1),p ] P [x(1),p ] (43) Note that the Jacobian matrices H and H must not be conused with the scalar Hamiltonian unctions H 1 and H 2 that are deined in Section II. Also, the generalized output vector unctions h and h must not be conused with the WGS-84 altitude unction h alt that is deined and used in Subsection III-C. D. Adjoint Calculation o Partial Derivatives o Outputs with Respect to Ionosphere Parameters This subsection presents the adjoint method or calculating the partial derivative o the initial output y and the terminal output y with respect to p i, the i th element o the ionosphere parameter vector p. The ormulas presented in this section have been derived by irst developing a directmethod solution using state transition matrix methods, as discussed briely in Subsection IV-A. The resulting solution was then manipulated to derive its adjoint equivalent. The necessary derivations are lengthy and have been omitted or the sake o brevity. The adjoint solution starts by solving a pair o linear matrix inal-value problem: dλ g (τ) = F T (τ)λ g (τ) dτ Λ g (1) = G T dλ h (τ) = F T (τ)λ h (τ) dτ Λ h (1) = H T (44) These two solutions are used to compute three vectors. The irst two are terminal group delay adjoint vectors, and the third projects the parameter p i s dierential equation right-hand-side sensitivities onto the terminal boundary condition: λ g = w + λ h = z + µ gi = µ hi = Λ T g(τ)u(τ)dτ Λ T h(τ)u(τ)dτ Λ T g(τ)u i (τ)dτ Λ T h(τ)u i (τ)dτ (45) Note: the adjoint method vectors λ g and λ h must not be conused with the eigenvalue λ that is used to calculate the Hamiltonian H 2 in Subsection II-B. Next the adjoint matrices o the initial and terminal boundary conditions are computed or the initial and terminal outputs: [ ] Λa Λ b = Λ a Λ b [ ] G T 1 [ ] Λ g () H T Λ h () w T λ T g z T λ T h (46) Finally, the initial and terminal output sensitivities are computed: y [ p i Λ T y = a Λ T ] [ ] [ ] a wi Λ T b Λ T + (47) b µ gi µ hi p i This is a very eicient computation because the only calculations that need to be re-done or a dierent parameter index i are those in the ourth line o Eq. (38), those in the last two lines o Eq. (45), and those in Eq. (47). E. Adjoint Calculation o Partial Derivatives o Outputs with Respect to Initial and Terminal Positions The calculation o output partial derivatives with respect to the initial and inal ray-path locations r and r reuses many o the quantities that are computed as part o the p i partial derivative calculation. The only new quantities are the partial derivatives o the initial and terminal boundary conditions with respect to the initial and terminal positions: G r = g r [x(),p,p, r] G r = g (48) r [x(1),p, r ] 12

13 The desired output partial derivatives are then y y [ r r Λ T y y = a Λ T ] [ ] a Gr Λ T b Λ T b G r r r (49) F. Proper Numerical Integration in Adjoint Calculations Care must be taken in the numerical integration o the matrix dierential equations in Eq. (44) and the integrals in Eq. (45). One might be tempted to calculate them using the Dormand-Prince 4/5 method or a similar method. The matrix inal value problems in Eq. (44) can be integrated backward in time rom the terminal condition at τ = 1. The integrals in Eq. (45) can be evaluated by simultaneously integrating the ollowing vector inal value problems: d λ g (τ) = Λ T dτ g(τ)u(τ) λ g (1) = w d λ h (τ) = Λ T dτ h(τ)u(τ) λ h (1) = z d µ gi (τ) = Λ T dτ g(τ)u i (τ) µ hi (1) = d µ hi (τ) = Λ T dτ h(τ)u i (τ) µ hi (1) = (5) backward in time rom τ = 1 and setting λ g = λ g (), λ h = λ h (), µ gi = µ gi (), and µ hi = µ hi (). The backward numerical integrations in the irst line o Eq. (44) and in the irst and ith lines o Eq. (5) must be carried out together as though a single large state were being integrated that consisted o the columns o Λ g (τ), the vector λ g, and the vector µ gi. I multiple µ gi are being evaluated or multiple parameter vector element indices i, then all o them must be included in the large state. Similarly, the backward numerical integrations in the third line o Eq. (44) and in the third and seventh lines o Eq. (5) must be carried out together as though a single large state were being integrated that consisted o the columns o Λ h (τ), the vector λ h, and all µ hi vectors. These backward integrations must not be carried out using the same Runge-Kutta numerical integration method as has been used to perorm orward integration o Eq. (3). Instead, a special adjoint o the particular Runge-Kutta method must be used. The adjoint o each Runge-Kutta step s sequence o orward integration ormulas is also a sequential calculation. It perorms reverse-time numerical integration using an altered Runge-Kutta computation recipe that is based on the orward method s same Runge- Kutta tableau. The calculations o the adjoint Runge-Kutta method are given in Eqs. (15) and (16) o [11]. The backward numerical integration o Eqs. (44) and (5) requires various values o F (τ), u(τ), and u i (τ) during each integration step. The values that must be used are computed during the orward numerical integration o Eq. (3) [11]. They can be stored or later use in the backward numerical integration o Eqs. (44) and (5). G. Ineiciency o Finite Dierence Sensitivity Calculations One might be tempted to avert the diiculty o implementing the adjoint partial derivative calculations o this section. An alternative would be to approximate the sensitivities by using inite dierences. The use o inite dierences in the present case is not recommended. There are two reasons or this recommendation. Both stem rom the act that each inite dierence calculation will require iterative re-solution o the TPBVP. Re-solution o the TPBVP is costly in terms o computation time, and the expensive TPBVP solution would need to be carried out on the order o n p times. Recall that n p is the number o elements in the ionosphere parameter vector p. Each TPBVP solution will require at least one expensive Newton iteration. The large number o TPBVP solutions will result in a computation time that scales like the ineicient direct sensitivities calculation method discussed in Subsection IV-B. Thereore, inite-dierence sensitivity calculation could prove very expensive computationally. A second reason to avoid inite-dierence calculations is the impact o possible residual errors in TPBVP solutions. Suppose that the Newton method has not ully converged or a reported TPBVP solution that is being used in a inite-dierence calculation. Then the small TPBVP solution error will contribute directly to an error in the inite-dierenced partial derivative approximation. I the inite-dierence increment is very small, then this error can be large. I the inite-dierence increments are large enough to make such errors negligible, then an increased amount o truncation error might adversely impact the inite-dierence sensitivity approximations. 13