# Heuristic relationship between number of satellites and GPS precision?

I'm a statistician conducting an analysis on animal movement based on data collected using ATS Wildlink 500 collars. As part of this analysis, I'm estimating GPS error assuming that the error in the measurement is normally distributed with the mean being the true location (x and y) with some standard deviation that is equal in the x and y dimension. I'm ignoring accuracy at this point, or rather, I'm assuming that bias is zero.

Obviously, there is way more to this, but as a first cut analysis, I'm happy with this assumption. However, the one thing that I want to capture at this stage is the likely degradation in precision based on the number of satellites associated with a given GPS fix. What I'm looking for is a simple relationship between the number of satellites and the standard deviation of the GPS error given that the fix was successful. Less than a minimum number of satellites (4) and the error is infinite (i.e. the fix fails). So really I'm just looking for the relationship when the number of satellites is above this number (say between 4 and 12).

Is it linear? Sigma = A + B * X, where is X is the number of satellites?

Exponential? Something like: Sigma = exp(A + B*X)

Unfortunately, I don't have raw data where I could investigate this relationship myself and I'm having trouble finding good resources online. So I'm going to have to fit this relationship blind to a certain extent. I'm just looking for a general guideline on how precision improves with increasing number of satellites.

• Besides the number of satellites, there are several other factors that can create errors and biases in GPS positions. The geometric position of the satellite constellation, ionospheric effects, signals bouncing on trees, buildings, etc. So such a relationship would be very diluted in other errors and biases. Commented Dec 3, 2018 at 20:58
• Have you tried logging positions from a known point for, say, several hours? What do the plot of those positions look like? Are they very dispersed? Also, for each point, do the devices log the number of satellites that were visible? Do your animals move a lot geographically? Or do they essentially remain in the same area for long periods? Commented Dec 3, 2018 at 21:04
• I am aware that there are many factors that can create error and bias. But I do not have data on these other factors, simply the number of satellites associated with each fix. I will be estimating the error from the data, what I am looking for is the functional form. Commented Dec 3, 2018 at 21:55
• i do not have access to the GPS collars, but I do have summary statistics from an experiment where the collars where compared to a higher accuracy handheld GPS. I do not have the raw data from those experiment which would break it down by the number of satellites. The collars were programmed to record a GPS fix every 15 minutes and each fix has the number of visible satellites recorded. The animals alternate between periods of movement and resting within a somewhat restricted territory. I have a model that distinguishes between movement and resting. Commented Dec 3, 2018 at 22:00
• Do your collars log an estimation of the accuracy of the positioning? Commented Dec 3, 2018 at 22:32

Regarding Position Errors

• The ATS Wildlink 500 ("ATSW500") outputs what is known as autonomous positions. The errors of autonomous positions are normally distributed if and only if (1) the device (actually the antenna) is static, i.e., it stays in exactly the same place, and (2) over a period of duration, e.g., several hours, and (3) you are logging ten data points (also called epochs) or more per minutes, and etc.

• All autonomous positions already come with "built-in" biases. In the case of ATSW500, it outputs the horizonal bias, i.e., the HDOP, which you can use to estimate the quality of the coordinate. The HDOP is preferred over the number of satellites as quality indicator. A four-satellite "fix" with low HDOP is better than a six-satellite "fix" with high HDOP.

• Note that ATSW500 also tell you the "dimension" of the fix, e.g., 2D or 3D. This is another quality indicator. A 3D "fix" is anytime better than a 2D "fix".

• Perhaps, by using the HDOP and "dimension", you could create an ordinal indicator of quality. Yes, this is not fanciful - just my thoughts. However, if you are still keen to explore some correlations, then it would most probably be linear (actually linearized). Thus far, I had not yet come across non-linear equations in the study/field of GNSS positioning.

Regarding Accuracy

• A common error in this forum is the failure of the Questioner to state the level of acceptable accuracy, e.g., deca-meter level, meter level, sub-meter level, and etc. For example, in tracking a storm, we don't need sub-meter level accuracy.

• I do not know what is the acceptable level of accuracy for the wildlife you are tracking - but if it is under 100 meter level, then the effects of ionosphere and multipath (eg., at the foot of a steep cliff, under tree canopy, and etc) are negligible.

Regarding ATSW500

• I obtained information of the ATSW500 outputs from the User Manual on its website.

• The manual also stated that the output height is not ellipsoidal height but mean sea level. If altitude is a concern, then you may wish to ask the manufacturer which vertical datum they applied, and the granularity.

All of this information is very helpful, but is not a direct answer to my question. That is "don't use the number of satellites, but rather some other metric."

• My answer could had been (in the context of your question, i.e., what you wanted to achieve) "There is no meaningful relationship between the number of satellites and GPS precision for autonomous positioning". It is short, but not helpful. Fact is - this is the GNSS engineering side of things.

• "All the other information" was a suggestion for helping you to achieve your objective, i.e., a possible way forward.

• ATSW500 outputs HDOP for every epoch (as stated in their User Manual). This is the single most important piece of information for your objective - not the number of satellites. Perhaps this old article may shed some light for you.

• In high-precision GNSS survey works, we want to have the largest number of satellites ("seen" by the antenna), not (mainly) because this gives better accuracy and precision, but so that the User or the intelligent post-processing software have options to exclude "problematic" satellites (or rather problematic observables recorded from the satellite(s) during the observation). Yes, large number of satellites isn't necessarily a good thing.

• However, the option to exclude satellites is not available in autonomous positioning performed by the firmware in ATSW500.

• So in my model (a hybrid hidden Markov/ dynamic linear model), I would be better off parameterizing the precision based off a metric other than the number of satellites? All of this information is very helpful, but is not a direct answer to my question. That is "don't use the number of satellites, but rather some other metric." I don't necessarily have an acceptable level of accuracy, rather what I'm looking to do is properly quantify the level of uncertainty in my estimated "true position." The data are what they are, I'm simply trying to do them justice in my model. Commented Dec 4, 2018 at 17:12
• I do not believe there can be a direct answer to your question based upon the quality of the receivers. They appear to be single frequency GPS units without the benefit of any SBAS corrections (WAAS, EGNOS) which would give at least some confidence that the positions are within a radius of several meters. PDOP, HDOP, VDOP and the other dilution of precision values can provide a very general idea as to the quality of position, with lower values equating to better positional confidence. For these units with HDOP less than 3, and five+ satellites you should be within 10 meters. Commented Dec 4, 2018 at 17:52

Typically, we do not use the number of satellites alone as a measure of precision or accuracy of GPS data. I do not know of any direct mathematical relationship that would link the two, and it would depend on the receiver as well. If you really wish to find a relationship, you would have to conduct a study with your collars and see if there is a correlation and statistically significant relationship, and find an approximate formula.

Also, keep in mind that while a GPS can pinpoint a location with a certain precision (consistency in its calculated location), it can still be off my several meters, so the positional accuracy is not the same. For example, if satellite signals tend to bounce on a cliff before reaching the receiver, or if there is greater ionospheric delay, you can get a positional error because of that, but you won't necessarily notice it in your data.

• This is getting very close to answer. Are you saying "In the GPS community there is no recognized relationship between the number of satellites and the standard deviation of positional errors"? If that is what you mean, than I will accept this as the answer. Commented Dec 4, 2018 at 17:19
• Well, to the best of my knowledge, no, there is no general recognized formula that could link the two, mainly because it would vary too much depending on the device, the geometry of the constellation, and other factors. The closest that you can get to such a relationship would be GDOP, but that takes into account the position of the satellites in the sky as well. Commented Dec 4, 2018 at 17:51