GPS Q&A
By Andrew Plackner

Q.There are so many ways to measure a GPS receiver's accuracy: RMS, CEP, 2DRMS, etc. What do these terms mean?
L.T. Rochester, NY

A.Understanding what each of these terms means is an important factor in evaluating GPS receiver performance. Perhaps the best way to explain these terms is by using a GPS receiver. This receiver is set to collect points over a period of time. After many positions have been gathered, these points, if plotted, would look like a scatter plot with points strewn about in a random manner. The variance of these positions may differ greatly from receiver to receiver. The resulting plot of points can be analyzed in a number of ways using the terms mentioned in your question. However, before we discuss measures of accuracy, let us first define two important concepts closely associated with a discussion such as this, accuracy and precision.
    While accuracy refers to the measured distance from a known point to an unknown point, precision pertains to the repeatability of a position. It is possible for a GPS receiver to be accurate but have very poor precision. Conversely, it is also possible for a receiver to be inaccurate but have reasonable precision. For better understanding, let us consider two GPS receivers, Receiver A and Receiver B. Each receiver has collected numerous measurements to calculate a final position. Receiver A has a corrected GPS position 100 centimeters from the true coordinate. The observations averaged to compute the position vary only a couple centimeters. The position collected by Receiver A has excellent precision, and poor accuracy. At the other end of the spectrum, Receiver B has a corrected position 25 centimeters from the true coordinate. However, the points used to calculate the final coordinate vary by as much as 50 centimeters. The resulting position from Receiver B exhibits good accuracy and rather poor precision. Excellent GPS receivers exhibit both good accuracy and precision. That is, the observations averaged to compute a final position vary little from one another, and the resulting corrected position is very close to the true coordinate.
    Now, back to the question at hand. As you might expect, these acronyms do not have identical meanings. However, what they do have in common is they are all a measure of statistical probability. Such an approach is made because of the myriad external errors that may affect GPS accuracy. Sources of error include excessive multipath, carrier phase cycle slips, spatial decorrelation over long baselines, insufficient signal strength, and poor field techniques. These factors are all part of GPS positioning. Fortunately, excellent GPS receivers and proper field training can eliminate much of the error associated with such phenomena. Some of the most common ways to evaluate the magnitude of error on the horizontal plane is by using the terms mentioned in your question.
    Root mean square (RMS) is the most common way to describe accuracy. RMS or HRMS (horizontal root mean square), equals the square root of the average of the squared errors along the x and y axes. While there is some variability in the derivation of the horizontal accuracy, 67% (one sigma) is the most common representation. In the "real world" this means that you can expect a 67% probability that your field results will fall within the stated distance from the actual coordinate.
    Similarly, two times the RMS results in a higher probability. A 2DRMS solution means that you can expect a 95% (two sigma) probability that your field results will fall within the stated distance from the true coordinate. It is important to note that here too, that 2DRMS is not always equivalent to a 95% probability level. As with RMS, 2DRMS may vary slightly. Circular error probable (CEP) is defined as the radius of a circle centered on the true location that collects 50% of the error distribution. It is the measure of the radius of a circle, where the true coordinates will fall 50% of the time. Similar to CEP is SEP, or spherical error probable. The main difference between CEP and SEP is that the latter is the measure of a sphere's radius (instead of a circle), centered at the true coordinate, containing 50% percent of the coordinates, this time in three-dimensions.
    When comparing these measurements strictly in terms of probability, variability exists from one term to the next. Considering an average measurement, CEP, and SEP (in three dimensions), allows for more error in the measurement of positions since its probability is only 50%. RMS is next at 67%, and 2DRMS is roughly 95%. All of these terms used to measure probability can be used evaluate GPS receiver performance. However, be sure to understand the differences that exist so that you can properly quantify and understand GPS receiver performance.

Q.What effect does the troposphere have on the GPS signal?
MLJ Panama City, Panama

A.The troposphere, along with the tropopause and the stratosphere, occupies the lower part of the atmosphere, from the surface to approximately 10-40 kilometers above the ground. At higher latitudes the troposphere is usually thinner, while near the equator it extends further toward the upper atmosphere. Derived from the Greek tropein, meaning to turn or change, this is a layer that contains all the weather we are familiar with: thunderstorms, rain, fog, hail, etcetera. This non-dispersive portion of the atmosphere affects both the carrier and code portion of the GPS frequency. In fact, both the L1 and L2 frequencies of the GPS signal are equally delayed since its impact, tropospheric refraction, does not depend upon frequency.
    Refraction is defined as the change of direction of light, heat, sound, etc. as it passes from one medium to another. The delay caused by this layer is a function of the tropospheric index, which is entirely dependent on temperature, pressure, and relative humidity. Modeling the troposphere helps to eliminate some of the error it introduces.
    The dry atmosphere contributes approximately 90% of the delay introduced by the troposphere. Fortunately, it can be modeled accurately. The wet atmosphere on the other hand is much harder to model due to the difficulty of predicting the local dispersion of water vapor. While this may be more difficult to model, the wet component represents approximately 10% of the total refraction.
    How does all this effect the GPS signal? Well, it depends on a lot of things including the particular makeup of the troposphere, and your location on the face of the earth. Left unchecked, the error for autonomous GPS positioning ranges from approximately two and a half meters in the zenith direction (directly overhead the observer) to about 25 meters at a 5 degree angle. The delay caused by the troposphere becomes increasingly uncorrelated as baseline distances on the surface increase. The delay will also increase for significant changes in air pressure. To mitigate such effects on the GPS signal a model is typically applied. For GIS/Mapping applications requiring no better than submeter accuracy, a model is most commonly applied behind the scenes along with the differential processing. For survey grade applications employing single or dual frequency receivers, the user is most commonly presented with a choice. Either apply the "standard" model for the troposphere, or make your own observations for temperature, barometric pressure, and humidity. Correctly applying a model to limit the error induced by the troposphere lessens the effect it has on GPS observables. Resulting errors are typically no greater than a few centimeters and are usually much less.

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