Bounding Higher Order Ionosphere Errors for the Dual Frequency GPS User

The advent of a second civil GPS frequency heralds a new phase of GPS performance. For single-frequency GPS users, the signal delay due to refraction through the ionosphere is the largest and most variable source of positioning error. Dual-frequency users take advantage of the dispersive nature of the ionosphere, combining the GPS observables to eliminate most of this error, on the order of meters, introduced into the pseudoranges. Thus it is with great anticipation that we await the improved accuracy afforded by L2, and for the aviation community, L5. However, with a second civil frequency, the ionospheric error will not disappear completely. Current techniques for measurement and removal of ionosphere delay using L1 and semi-codeless tracking of L2 typically assume a first-order approximation of the index of refraction in the ionosphere. This approximation results in a range delay inversely proportional to the square of the signal frequency, and equal and opposite to the phase advance. Once this correction is made, higher order terms are then the largest ionosphere error. This paper examines the magnitude of these higher order ionospheric error terms. Previous analyses from the early 1990s used simulations to show that typical higher-order terms should be “much less than 1% of the first order term at GPS frequencies” (Klobuchar 1996). We verify such values with L1-L2 dual-frequency GPS data from the current solar cycle, made available by the Wide Area Augmentation System (WAAS) network. We compare the magnitude of the second and third order terms. We assume the International Geomagnetic Reference Field IGRF-10 (10 generation, created 2005) and take its value at 350 km shell height. For the third order term involving the square of the electron density, we use a technique from Hartmann and Leitinger (1984) to approximate the ionosphere vertical profile with a maximum density Nm and a shape factor. We choose a shell model with 100 km thickness and compute the uniform density consistent with WAAS equivalent vertical delay measurements as Nm, giving a shape factor of one. We find that the Brunner & Gu (1991) model is the simplest one that neglects the sub-millimeter terms, and use this to compute the higher order group and phase errors that occur from the use of the observable that is ionosphere-free to first order (FOIF). During the most active of these days, when ionospheric storms may introduce slant range delays at L1 as high as one hundred meters, the higher order group errors in the FOIF combination can be tens of centimeters. Moreover, the group and phase errors are no longer equal and opposite, so these errors accumulate in carrier smoothing of the FOIF code observable. We show the errors in the carrier-smoothed code are due to higher order group errors and, to a lesser extent, to higher order phase rate errors. For many applications, this residual error is sufficiently small as to be neglected. However, such errors can impact geodetic applications as well as the error budgets of Augmentation Systems providing Category III precision approach. INTRODUCTION The ionosphere is a weakly-ionized plasma layer of the upper atmosphere from about 60 – 1000 km. As a dispersive medium, the ionosphere advances the phase and delays the code of the GPS ranging signals in a frequency-dependent way as they travel through it. These result in errors in the code P1 and P2 and carrier phase measurements L1 and L2 made by the user: