GPS Q&A

Q Can you tell me what RMS means, and how it applies to GPS accuracy? I have noticed this term on some specification sheets for GPS products.

A Specification sheets for GPS receiver products will invariably quote an accuracy statement. It can be easy to misinterpret this statement, especially if it is not qualified with an indicator of RMS. RMS, or Root Mean Square, describes the error distribution of a given statistic, and its use is important in understanding a specification statement.

Root Mean Square Errors
If you have ever seen a plot of GPS positions over time from a stationary GPS antenna, you know that the positions are scattered over an area. When GPS manufacturers characterize their GPS equipment, they use these "scatterplots" to determine the equipment's accuracy.
    A typical scatterplot will have a circular shape with a central cluster of positions. Toward the edge of the circle, the density of positions gets more sparse. When characterizing GPS equipment, many scatterplots are generated and analyzed using a wide variety of GPS environments. This makes it virtually impossible to define an absolute radius of spread, which would be the accuracy of the GPS equipment.
    Instead, statistical methods are used to calculate the error distribution of the positions from the various scatterplots and an RMS error value is generated. The value quoted for RMS is the radius within which approximately 63% of the positions fall. (The actual percentage may vary slightly, but we can assume it is 63% for the purposes of the discussion.) For example, if a specification sheet claims that the equipment it advertises has an RMS level of 1-meter (m) on a position-by-position basis, then statistically speaking, a GPS position will be accurate to within 1m of truth 63% of the time. This can also be referred to as the instantaneous RMS error.

Confidence Levels
Another way of referring to RMS errors is by confidence level. For example, an accuracy specification of 1m RMS means that you are 63% confident of obtaining this accuracy for any given position. However, higher levels of confidence can be used. Some specifications use 2D RMS, which is two times the RMS accuracy value. Imagine a scatterplot with many positions and a circle drawn at a 1m radius encompassing 63% of the positions. The radius of this circle is the RMS accuracy. Now draw a circle that is twice that radius; 2m in this case. This circle will encompass somewhere around 95%, of the positions. You can see here the relationship between levels of accuracy (RMS or 2D RMS) and the matching levels of confidence. In the example above you can be 95% confident of getting 2m, but only 63% confident of getting 1m.
    Occasionally, you may see a specification quoted as CEP. This acronym means Circular Error Probable, but you can think of it as another form of RMS at about the 50% level, meaning statistically that 50% of positions should meet or exceed this level of accuracy.

Horizontal Versus Vertical Accuracy
Although many data collection applications are two-dimensional, the third dimension-or height component-is becoming increasingly important as project requirements become more complex. Almost always, the accuracy statement on a specification document will only be for horizontal positions, i.e., in the XY plane. In reality, a scatterplot is three-dimensional because GPS positions have an X, Y, and a Z component, and it is not a circle which is formed around the positions but an ellipsoid. Quite consistently, the ellipsoid formed from GPS positions is about twice as long in the Z axis as it is in the XY plane. So, as a rule of thumb, you can double the accuracy figure to establish the accuracy of a piece of GPS equipment in terms of height. Thus, a receiver with a 1m horizontal accuracy (RMS) will have a 2m vertical accuracy.
    Very occasionally a scatterplot will even show this ellipsoidal bias in just the horizontal plane. If there is a marked difference in the east-west and north-south axes, then two RMS values may be published, one for each axis. These can be taken together (along with the 2x stretching in the vertical) to form a very specific error ellipsoid.

"Accuracy" Versus "Precision"
A brief discussion of the terms "accuracy" and "precision" is appropriate here, because you may see both printed on a specification sheet. To many people they appear synonymous, and on a spec sheet they may be used interchangeably. However, it is worth understanding the subtlety between these two words.
    "Accuracy" is an absolute term. For example, 'This GPS position is accurate to within 1 meter.' This implies a knowledge of the true real-world position, and thus the term is often used when referring to permanently mounted reference stations.
    "Precision" on the other hand is considered a relative term. You can have a very precise set of GPS data in relation to itself (for example, nice straight roads, poles aligned evenly and neatly along the curb), but there is no real way of telling how well these positions overlay in reality.
    The best way to convert "precise" data to "accurate" data is to record the location of some known points during the data collection session. By comparing these GPS points with their published equivalents, it is possible to determine how accurate your data really is, and even adjust it if necessary. In fact, the term "network adjustment" used by surveyors is exactly this; adjusting a network of precise GPS positions to accurately fit an existing control network.

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