ADOP in closed form for a hierarchy of multi-frequency single-baseline GNSS models

Abstract

Successful carrier phase ambiguity resolution is the key to high-precision positioning with Global Navigation Satellite Systems (GNSS). The ambiguity dilution of precision (ADOP) is a well-known scalar measure which can be used to infer the strength of the GNSS model for carrier phase ambiguity resolution. In this contribution we present analytical closed-form expressions for the ADOP. This will be done for a whole class of different multifrequency single baseline models. These models include the geometry-fixed, the geometry-free and the geometry-based models, respectively. And within the class of geometry-based models, we discriminate between short and long observation time spans, and between stationary and moving receivers. The easy-to-use ADOP expressions can be applied to infer the contribution of various GNSS model factors. They comprise, for instance, the type, the number and the precision of the GNSS observations, the number and selection of frequencies, the presence of atmospheric disturbances, the length of the observation time span and the length of the baseline.

The Future of Single-Frequency Integer Ambiguity Resolution

“…It is possible to derive closed-form expressions for ADOP. In Odijk and Teunissen (2008) this was done for a hierarchy of multi-frequency single-baseline GNSS models. The closed-form expressions give a clear insight into how and to what extent the various factors of the underlying GNSS model contribute to the overall AR performance, see Odijk and Teunissen (2007).…”

“…The closed-form expressions give a clear insight into how and to what extent the various factors of the underlying GNSS model contribute to the overall AR performance, see Odijk and Teunissen (2007). The closed-form expression for the ADOP of the single-frequency model corresponding to a moving receiver covering a short time span (no change in satellite geometry) can be derived as (see table 8 in Odijk and Teunissen (2008), use j D 1): The ionosphere-weighted model, see e.g. Odijk (2002), is used where a priori information on the ionosphere delays is used in the form of ionosphere observables with standard deviation à depending on the baseline length.…”

A dense reference network for mass-market centimeter-accurate positioning

“…This model, presented in [12] and [14], relates ADOP to the following parameters: the standard deviation of ionospheric correction errors σ ι , the number of visible satellites m, the standard deviation of undifferenced carrier-and codephase measurement errors, σ φ and σ ρ , respectively, (including multipath-induced errors), a satellite geometry factor f g , the number p of free parameters to be estimated (p = 3 for negligible tropospheric error, p = 4 to estimate a single additional tropospheric parameter), and the number of carrier frequencies broadcast by each of the m satellites (e.g., 1, 2, or 3) along with each carrier's wavelength. The model is highly accurate for single-epoch AR, but only approximate for multiple epochs, with accuracy degrading as the data interval lengthens.…”

“…For singlefrequency application of the analytical AR model in [12], exploited in the foregoing section, one has only to substitute σ ν for σ ι to properly account for the additional effects of residual tropospheric error and network-side multipath errors in the corrections data. Thus, on the horizontal axis of Fig.…”

“…Prior work has established an analytical connection between uncertainty in the ionospheric corrections, denoted σ ι , and TAR, beginning with the introduction of the ionosphereweighted model in [11] and culminating in the analytical formulations for so-called Ambiguity Dilution of Precision (ADOP) in [12]. ADOP can be used to accurately predict the probability P (ẑ = z) that the vector of estimated integer ambiguitiesẑ is equal to the vector of true ambiguities z. TAR can be defined as the time required for P (ẑ = z) to rise above a specified value (e.g., 0.99), or, equivalently, for ADOP to fall below a specified value (e.g., 0.12 cycles).…”