# 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__