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

Methods are developed to determine the refracted propagation paths of high-frequency RF signals in the ionosphere and to determine the sensitivities of these paths to changes of the input model parameters. These techniques are being developed to support the assimilation of data from mono-static and multi-static ionosondes with the goal of improving parameterized estimates of ionosphere electron density profiles. An additional application area is that of navigation using signals from a ground-based network of high-frequency 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 carefully designed first guesses of the wave vector’s initial aiming angles and terminal group delay. Partial derivative sensitivities of the ray-path solution are calculated using the adjoint of the two-point boundary value problem. This approach speeds the calculations when the partial derivatives of many ionosphere model parameters need to be computed. The new algorithm has been applied successfully to determine the paths of O-mode and X-mode radio waves between known transmitter and receiver locations and to spitze reflection points. Spitze singularities pose no difficulties for the new algorithm because it uses a Hamiltonian raypath formulation that remains non-singular at a spitze. INTRODUCTION Several areas of current scientific and engineering interest require an ability to calculate the ray paths of high-frequency (HF) radio signals through an ionosphere model. They also require an ability to determine how model changes affect the ray paths. One application is the fusion of ionosonde and GPS data to develop a local ionosphere model [1]. Another is the joint estimation of receiver position and ionosphere model corrections based solely on the observables of signals received from a ground-based network of HF beacons [2]. A third application is the assimilation of data from a network of HF beacons/receivers into a regional ionosphere model [3]. The latter two application areas are illustrated in Fig. 1. Two high-frequency beacons are shown as blue dots, and a receiver is shown as a red dot. A third beacon is out of view over the horizon in eastern Canada. Three refracting ray paths are shown between the 3 beacons and the receiver, one in green, one in purple, and one in tan. Two of the ray paths undergo multiple hops, but the purple one undergoes just a single hop between the transmitter and the receiver. Reference [2] proposes to use observables from these ray paths to estimate the receiver position, the receiver clock offset, and corrections to a parameterized model of the ionosphere electron density profile Ne(~r;p), where ~r is the Cartesian position vector along the ray path and p is a vector of parameters that characterize the profile. 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 of HF beacons/receivers constitutes a sort of multi-static ionosonde. This paper has two main objectives. The first one is to calculate the observables of a HF ray path that refracts through the ionosphere by solving the ray-tracing equations. These observables include the group delay, the carrier phase, and the arrival and departure directions of the signal at the two ends of the ray path. Two types of ray paths are considered. One type undergoes total reflection in the ionosphere so that the transmitter and the receiver are collocated. This is the ionosonde rayCopyright © 2016 by Mark L. Psiaki. All rights reserved. Preprint from ION GNSS+ 2016 with corrections not found in proceedings version. Fig. 1: High-Frequency beacons, a receiver, and representative ray paths that refract off of the ionosphere. path problem. The second type of ray path is a point-topoint path from a known transmitter location to a different known receiver location. Ray paths of this second type can be concatenated to produce multi-hop paths. The explicit calculation of multi-hop paths is beyond the scope of this paper, except that this paper’s developments for single-hop paths will provide all of the needed outputs for use in a concatenated multi-hop calculation. This paper’s second main objective is to develop a means of computing the partial derivative sensitivities of the ray-path observables to problem inputs. For ionosonde reflection ray paths, the needed sensitivities are the partial derivatives with respect to the elements of p, which define the ionosphere electron density profile Ne(~r;p). For point-to-point ray paths, additional calculations are developed for the partial derivative sensitivities with respect to the transmitter and receiver locations. This paper makes four contributions to the technique of RF ray-tracing. The first is a modified pair of Hamiltonians, different from those given in [4], for use in defining the Hamiltonian differential equations of the ray path. The second contribution is a spline method for transitioning from the differential equations that are based on the first Hamiltonian to the differential equations that are based on the second Hamiltonian. The first Hamiltonian applies near free space, and the second applies in regions of high electron density, i.e., near a reflection point/spitze. The third contribution is a robust solver for 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 final group delay that cause the ray-path to satisfy the terminal boundary conditions. This ray-path solver seeds Newton’s method with good first guesses that it generates using a simplified ray-path model. The fourth contribution is an adjoint-based method for calculating the sensitivities of various ray-path observables to problem parameters. The calculated sensitivities are partial derivatives of the TPBVP solution observables taken with respect to input quantities that define the TPBVP. The remainder of this paper is divided into 4 main sections plus conclusions. Section II defines the 2 new Hamiltonians, and it uses them to develop the ray-tracing differential equations. This section also presents the two sets of boundary conditions for the two classes of ray-tracing problems that are dealt with in this paper. Section III describes the solution method for the ray-tracing nonlinear TPBVP. It develops a combined shooting-method/Newton’smethod. It includes a technique for generating good first guesses to seed the shooting/Newton algorithm. Section IV derives methods for determining the partial derivative sensitivities of the ray-tracing solution and its observables with respect to problem inputs that include ionosphere parameters and the initial and final locations of 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 RAYTRACING PROBLEM This section formulates the differential equations and the boundary conditions of two ray-tracing problems. One is an ionosonde reflection problem, and the other is a pointto-point problem.