Error Manifolds

Circular Error Probable (CEP)

Let x and y be zero mean Gaussian, each with standard deviation s_x = s_y

Let z = radial error. Then:

z has a Raleigh distribution:

Linear Error Probable (LEP)

x is a zero mean Gaussian random variable with standard deviation s_x

Spherical Error Probable (SEP)

Often it is useful to characterize navigation accuracy in terms of spherical errors. In satellite navigation systems such as GPS and GLONASS, horizontal accuracy is usually much better than vertical accuracy because vertical information is developed from satellites at high elevation angles. Typically, the highest elevation satellite is only at 45 to 50 degree elevation whereas there are a multitude of satellites at lower elevation angles. This fact is reflected in HDOP and VDOP values where VDOP is usually about 1.5 to 2 times as large. While it is possible to come up with an analytic formula for SEP; the formulas are sufficiently difficult to evaluate so as to be useless. Here, we will take a more pragmatic approach and develop simple formulas based on Monte Carlo simulation results.

Figure S1 plots normalized spherical error as a function of probability for several VDOP/HDOP values based on simulation.To compute SEP from this chart:

SE_norm is based on VDOP/HDOP and the appropriate probability value. If you are interested in computing a 95% confidence radius, use the corresponding value from the chart. 95% of the navigation solutions will have spherical error less than or equal the computed SEP. Sigma_radial is simply the standard deviation of bias type errors in measurement of ranges to individual satellites. Note that this formula ignores navigation filter residual errors due to thermal noise and dynamics. These errors are usually a small component of the overall error budget except in high performance differential systems.

Figure S1: Normalized Spherical Error As A Function of Probability

It turns out that SEP can be fairly accurately computed using the formula:

where CEP and LEP values are computed at corresponding confidence values. Figure S2 plots simulated SEP and computed SEP based on the above formula at VDOP/HDOP=2.

Figure S2: Normalized Spherical Error As A Function of Probability for VDOP/HDOP = 2(Exact & Formula)

Dividing the formula SEP by actual SEP at several VDOP/HDOP ratios, we obtain Figure S3. In the neighbourhood of 50% confidence radii, we see that the formula works very well while at greater confidence radii, we see that the simple formula predicts SEP radii somewhat larger than they actually are. Such is the price for simplicity. In my gps_accuracy.xls spreadsheet, I use the simple formula for 50% radii directly, and at 95% radii, I divide by a correction term (1.415-0.14487(VDOP/HDOP) ) so as to obtain + 2% accuracy in SEP calculations.

Figure S3: Ratio of Formula SEP to Actual SEP As A Function of Probability for VDOP/HDOP = {1..3}