## 1. Introduction

[2] This paper investigates the capability of a system for determining ionospheric specification from two-frequency GPS receivers on the ground. Specification consists of the distribution of electron density in latitude, longitude, and height. Satisfactory system performance requires accuracy and timely distribution of the specification. We previously developed a method of processing GPS receiver data [*Reilly and Singh*, 2001] that uses the electron density model in the Ionosphere and Troposphere Raytrace Model (ITRAY), our upgrade of the Raytrace/ICED-Bent-Gallagher (RIBG) model [*Reilly*, 1993]. In this method, least squares analysis provides both a single effective sunspot number driver of the height profile model and hardware differential biases. The data consist of 2 hours of shifted differential phase pseudorange (SDPP) (L1–L2) data in total electron content units (TECU) (one TECU = 10^{16} electrons per cubic meter) on 5 satellite receiver paths. The data interval is 30 s. Two hours of data provide enough ionospheric variation to obtain ionospheric model driving parameter and relative hardware differential bias (RDB) solutions. SDPP data traditionally serve as a proxy for group path length without the effects of multipath and without the free space range contribution. We showed evidence from sounder measurements that ITRAY, updated by this method, could predict *f*_{o}*F*_{2} within a few MHz. In this paper, we attempt to exploit the flexibility of the EDP model in ITRAY by varying four of its driving parameters independently in order to obtain a better fit of the GPS data. The four driving parameters are: (1) SF2, which determines the value of the maximum plasma frequency *f*_{o}*F*_{2}; (2) SM3, which determines the value of M3000 (multiplying factor of *f*_{o}*F*_{2} that gives the maximum usable frequency (MUF) on a 3000 km path), which determines *h*_{m}*F*_{2} (height at maximum plasma frequency) in the ionospheric model; (3) SWDTH, which determines the width of the F2 Chapman layer in the model; and (4) CFAC, which affects how fast the EDP drops off with height above *h*_{m}*F*_{2}. A fifth driving parameter of the model is *K*_{p}, the 3-hourly planetary magnetic activity index, which we determine separately from external geomagnetic data. The conditions SF2 = SM3 = SWDTH and CFAC = 0.86 were previously used to constrain these parameters.

[3] The theory for fitting the model to the data is similar to before [*Reilly and Singh*, 2001] except that we now work with differences of SDDP data between the satellites, or relative SDDP data, in order to remove the influence of receiver clock drift. In our previous paper, we assumed that hardware differential biases between each satellite and the receiver would remain constant over the 2-hour period. Now that we work with relative SDDP data between satellites, the assumption of constant RDBs is substantially more reliable. Hence the number of unknown model parameters for five satellites is now eight: four driving parameters of the ionospheric model and four RDBs. We obtained nearly identical answers from this and the previous method for many cases, thus confirming the method. A discrepancy found in a few cases was apparently due to receiver clock drift.

[4] The method we choose to obtain the multiparameter fit of relative SDDP data is the Levenberg-Marquardt (LM) method, as implemented and explained in the work of *Press et al.* [1992]. We adapt this implementation to our problem as follows. First, obtain the solution for the single sunspot number driver in the above default condition. We later refer to this as the *F* solution. This takes less than 1 min on a typical personal computer (PC) for a wide range of initial guesses of starting sunspot number. The *F* solution could be the starting guess for the LM method, which finds model parameter solutions that minimize the value of a chi-square merit function (chisq), given by

where *x*_{i} is an index parameter that runs over the satellite pairs and the time, *y*_{i} represents a particular one of *N* relative SDDP data points, and *y*(*x*_{i}, *a*) represents the model calculation of this datum as a function of the 8-element model parameter vector *a*. We assume the data to have equal weights. Unfortunately, the *F* solution guess sometimes converges to a local minimum of chisq, which is different from the global minimum. We need the starting guess close enough to the global minimum solution in order to avoid this. Hence we start with a grid of driving parameters centered on the preceding *F* solution guess. This grid presently consists of a range of values for each of SF2, SM3, SWDTH, and CFAC. For each value in this grid we calculate RDB values that minimize chisq. The overall minimum chisq grid point is used as the new starting guess for the LM method. We refer to this later as the LMG solution. Convergence of LMG takes about 50–150 times longer than *F* since several tens of iterations and many more ionospheric model driving parameter interpolations are involved.

[5] The next section calculates the results of analysis of Westford GPS data for 6 dates in 2000 and 2001. The Westford receiver is located at 42.613°N–71.493°E. We compare updated ITRAY model predictions of EDPs with available EDP data from the Millstone Hill incoherent scatter radar data and Digisonde data over the 2-hour GPS data processing period. Millstone Hill is located at 42.620°N, –71.492°E. This is a severe test of the ionospheric model since available incoherent scatter radar (ISR) data concentrate in the late afternoon to evening hours, when the ionosphere varies rapidly. We use a single set of driving parameter solutions for the predictions, thus relying only on the time dependence within the ionospheric model.