| 10 Dec 2025
An update to the expression of atmospheric refractivity for GNSS signals
Abstract. This study revisits previous formulations of atmospheric refractivity at L-band frequencies, focusing on signals from Global Navigation Satellite Systems (GNSS). A refined model expression is proposed as a function of air density, temperature, and composition, evaluated using a comprehensive set of existing laboratory and atmospheric measurements. The key measurements that most affect the final accuracy are identified, establishing traceable error bounds and indicating where further experimental work could confirm or improve the model.
Recent studies on the use of large volumes of GNSS radio occultation (GNSSRO) observations in Numerical Weather Prediction (NWP) show that the precise formulation of refractivity becomes increasingly critical as data volumes grow. Although the revision is modest, its impact lies within the range where NWP sensitivity becomes non-negligible.
Compared to earlier work, this study (1) incorporates updated fundamental measurements, (2) accounts for the small but measurable variability in atmospheric composition, mainly increasing CO2 and decreasing O2, emphasizing that refractivity traceability is composition-dependent, and (3) extends the model to include hydrometeors. A simplified formulation based on hydrometeor oblateness is proposed, suitable for NWP applications where only limited hydrometeor information is available. Nonspherical hydrometeors tend to align during fall, introducing weak birefringence that can be detected during GNSS occultations with dual-polarization receivers.
The resulting refractivity expression is presented as a function of air density, temperature, moisture, and composition, and (using a simplified model of atmospheric evolution) also as a function of density, temperature, moisture, and time.
General Comments:
The paper is a welcome addition to the literature. It builds on a previous work, Aparicio and Laroche (2011) (AL11), and adds significant new elements including accounting for variation in atmospheric composition (CO2, O2) over time, and the refractivity contribution of hydrometeors. For the first time to this reviewer’s knowledge, the paper discusses the birefringence of the atmosphere in the presence of hydrometeors and in the context of radio occultation (RO), providing useful estimates of the how the propagation path varies depending on polarization. This paper provides traceability from the uncertainties of the fundamental measurements of microphysical properties to uncertainty of the expression for refractivity, which is a significant new contribution. The paper is clear and concise.
The target accuracy for the refractivity expression is 0.01%, the same as the accuracy claim in AL11. A careful analysis establishes that this target accuracy is achieved, even with the additional effects considered such as composition and hydrometeors. This reviewer suggests updating the abstract with the stated accuracy goal and that it is achieved with the revised expressions.
Specific Comments:
Abstract: as noted above, the abstract would be improved if the stated accuracy goal of 0.01% is mentioned. Also useful to mention is that the new analysis is consistent with this goal.
Line 66: this paragraph suggests that biases detected by NWP centers using ROMEX data are solely due to the refractivity formulation used. While the refractivity formula probably plays a role, is it justified to ascribe all the biases seen in ROMEX experiments to that formula? I believe this is not the generally accepted view.
Line 169: this formula references AL11 which does not include the gas phase term $\epsilon^r_g$. How is it decided which of the epsilons in AL11 equation (2) becomes the term for the gas phase?
Line 274: it would be very useful to have a quantitative upper limit for hydrometeor concentration consistent with the accuracy goals of the paper. Maximum values are listed near line 550. Presumably under conditions when these maxima are reached, the formula does not achieve 0.01% accuracy? If possible, an upper limit of hydrometeor concentrations consistent with 0.01% accuracy would be useful to have in the paper.
Line 288: “canting” is a somewhat specialized term in this context so please briefly define it.
Line 567: 200 m difference in altitude is significant, especially since papers have claimed to determine PBL height to approximately this level of precision. Could the birefringence affect estimates of PBL height?
Line 570: the descent rate slows considerably in the lower troposphere. Is this the appropriate descent rate in the PBL?
Line 578: please be explicit here that “useful” means accurate to 0.01%, if that is what is meant.
Line 724: this paragraph is a little confusing. The threshold of 0.1% is met? (rather than exceeded?). The second sentence is hard to understand. What is meant by “margin”? Existing formulations certainly do not meet 0.01%, even in the number of significant digits retained for the constants (in some cases).
Technical Corrections:
Line 177: remove extra “of”
Line 385: remove extra “is”
Line 431: should this be Eq (18)?
Line 576: remove extra “have”
Line 596: should be (28)-(38)?
Line 635: should be “dependent”, and remove extra “be”
Line 640: remove “po”
Line 650: should be “carries”