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19 Jan 2022
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STATISTICAL DATA DESCRIPTIONS

This page provides a description of each data item listed on the statistics pages. The Root Mean Square (RMS) values are calculated using the equations in the sections below. The 50% and 95% values are Percentiles. A Percentile provides information about how the data items are spread over the interval from the smallest value in a data set to the largest value in a data set. Details on each method are included in the appropriate section below.

GIANT, the GPSOC analysis software, derives regional Prediction and Post-Assessment data by calculating GPS accuracy at individual grid points spaced at regular intervals throughout the region of interest. Statistical data (RMS, 50%, 95%) for each grid point are then compiled into a single value representative of the region as a whole.


Global/Regional Position Error Post-Assessments

Post-Assessment data are based on ACTUAL position errors as observed by the GPS Master Control Station’s global network of GPS monitor stations. These values are intended to provide a quick-look post mission assessment capability to users. The daily products for Post-Assessment shown in Figure 1 are the horizontal, vertical and position statistics, the max position error and the RMS Signal-in-Space user range error.

figure 1
Figure 1

The Horizontal Position Error 50th Percentile (50%) is commonly referred to as Circular Error Probable (CEP). Likewise, the 3-D Position Error 50th Percentile is commonly referred to as Spherical Error Probable.


Global/Regional Dilution of Precision (DOP) Predictions

For Predictions, the daily products shown in Figure 2 for the regions are HDOP, VDOP and PDOP statistics, the 95th Position Error and the max PDOP for the day.

figure 2
Figure 2

The dilution of precision (DOP) is a purely geometric factor that takes the unit vectors from a receiver’s location to each of the GPS satellites in use and outputs a dimensionless geometric factor to multiply the pseudorange error in order to estimate GPS accuracy. Because DOP is a multiplier, it can have a significant impact to users of GPS. This can be seen in the equation for the accuracy of a GPS guided munition

equation1

UERE is the User Equivalent Range Error representing the total system ranging error. UERE comprises the User Equipment Error (UEE) and the Signal-In-Space (SIS) range error components

equation1

Foregoing a detailed derivation of DOP, the calculation is straight forward beginning with the unit vectors from the grid point of interest to the satellites in use

equation1

DOP is then calculated from the terms in the matrix

equation1

With the assumption that the x-axis points east, the y-axis points north and the z-axis points up, the DOP relationships are depicted as

dop relationship

An example output for the CONUS region is given in the below figure for the Best 4 satellite solution. The calculations for these statistics begins with GIANT calculating the average RMS DOP values for all the grid points over the entire region at one minute intervals over a 24 hour time period.


example rms
Figure 1

example_rms2
Figure 2

figure 2

In these RMS DOP equations, P is the number of time increments in the day and N is the number of grid points in the region. Calculating the other percentile values (50%, 95%) are done with the below percentile algorithm. Below depicts a rough guide for DOP performance.

dop good

Note that the percentile algorithm used for predicting DOP statistics and for post-assessment statistics is different than the empirically derived scale factors used for the 95% position error predictions. This prediction scales the RMS calculated value by an empirically derived scale factor to get the 95% position error. The conversion is

95% Position Error 2.035 * RMS

Percentile Algorithm

The percentile calculations (50%, 95%) for DOP predictions (Figure 1) and the Post-Assessment statistics (Figure 2) are calculated analogously to the sorted Histogram in Excel with the Cumulative Percentage. For the DOP predictions, GIANT uses 892 bins between 0 and 10 and counts up the number of DOP measurements in each bin until it gets to ( %P)*(N+P). As an example, if we were calculating the 95%, then the count would be (0.95)*(N+P).

A simple example with VDOP only can demonstrate the algorithm with these steps

1. Create 892 bins between 0 and 10. This works out to a bin width of 10/892 = 0.0112108.

figure 3
Figure 3

Then select the VDOP values for the histogram. Next go to Tools->Data Analysis.

figure 4
Figure 4

The Data Analysis window will pop up. Choose the Histogram option as shown in Figure 5.

figure 5
Figure 5

The next step is to input the data. Put the VDOP values into the Input Range and the Bins in to the Bin Range. Also make sure the Cumulative Percentage is checked as shown in Figure 6.

figure 6
Figure 6

Although the output Histogram sheet gives us the percentile answers in the Cumulative % column (Figure 7), we create another column to demonstrate how the algorithm counts the data points to get the percentile. The data point we count up to for the 95% is 0.95*(N+P) = 0.95*5760 = 5472. The Count column is simply just a sum of the frequency column up to the current cell. As an example, the cell D4 =SUM($B$2:B4).

figure 7
Figure 7

Next we go down and find which bin (Column A) would have the 5472 data point (Column D), which corresponds to 95th percentile. As shown in Figure 7, we see this value is 2.56, which contains data points 5471 – 5480. If we look at Column C, we see that Excel has already calculated this with the Cumulative percentage. Note that the value calculated here is a little off from the reported 95% VDOP number in Figure 1. This is because the simplified example only used 4 grid points for CONUS, while the daily product uses 7865 grid points.