As everyone knows, a GPS receiver must have a minimum of four satellites visible in order to determine the position of the receiver. The observation(s) are the travel time of the C/A code from the satellite to the receiver. The travel time, Dt, is multiplied by the speed of light to get the pseudo range.
Pseudo Range = Dt * Speed of Light
The receiver uses an “average value” for the speed of light, and the value of Dt may contain clock errors. That is why we call the range determined pseudo range; pseudo means false.
Figure 1 shows a GPS receiver with four visible satellites. For simplicity, assume the position of each satellite is known “exactly” at a particular instant of time (epoch), and the positions are represented as an X, Y, Z in the WGS 84 coordinate system. Assume now, at that particular epoch, a pseudo range has been determined from the receiver to each satellite. The receiver will solve for the position of the receiver as an X, Y, Z coordinate in the WGS 84 coordinate system. To make it convenient for the user, the receiver converts the X, Y, Z position to latitude, longitude and ellipsoid height.
Back to the solution for the receiver position. In addition to solving for the position X, Y, Z, the receiver also solves for the “clock offset.” That’s why it’s necessary to observe a minimum of four satellites. Because the pseudo ranges have errors, and the clock has an error, the position of the receiver will have an error in each component of its position. The receiver position is determined using the theory of least squares, and a variance-covariance matrix is generated for the receiver position and the clock offset.
Everybody uses matrices to solve least squares problems. (If you, GPS Observer readers, are interested I could write a two-part article to bring you up-to-date on that subject.) For this article, however, let me give the results of the least squares adjustment for the receiver position and the variance-covariance matrix.
The matrix of normal equations is
N = AT (-1 A,
where A is a matrix of partial derivatives and (-1 is the inverse of the measurement variance-covariance matrix. In other words, the diagonal elements of ( would be the variances of the measurements, which are pseudo ranges; ( is a 4 x 4 matrix for four visible satellites. If an accurate estimate of the pseudo ranges is known, the variance of the station position components and clock offset will be reliable. In order to solve for the receiver position (X, Y, Z) and clock delay, the normal equations must be inverted,
N-1 = ( AT (-1 A )-1.
This is also a 4 x 4 matrix, and the diagonal elements are the variances of X, Y, Z and the clock offset. Let’s refer to them as s2X, s2Y, s2Z, s2Ec. For the purist reading this column, I’m assuming the variance of unit weight is 1.
How does this relate to Dilution of Precision? The trace of N-1 is the sum of the diagonal elements (s2X + s2Y + s2Z + s2Ec). This is one number, and although I shortened the development, it’s a unitless number (the units are actually m/m), and it’s an indicator of the geometrical strength of the solution. Manufacturers of GPS receivers have simplified the calculations by saying that (-1 is the identity matrix (one is down the diagonal, everything else zeros). By doing this, we can define the Geometric Dilution of Precision (GDOP) as
GDOP = (trace ( AT A )-1 )1/2.
As most people know, the ideal solution is to have a GDOP value less than 5; the smaller the better. If you are concerned only about position and not the clock, the term s2Ec can be eliminated. This is defined as a Position Dilution of Precision (PDOP),
PDOP = ( s2X + s2Y + s2Z )1/2.
We can now mention the common description of GDOP and PDOP. If the satellites are close together, the geometry isn’t the best and the DOP value may be greater than 5. If the satellites are spread out, in all four quadrants, you will have a DOP less than 5 and a better solution for the position.
As stated earlier, a GPS receiver can calculate DOP. In addition, mission planning can generate tables of PDOP and plots of PDOP and GDOP for any location at any time if a current ephemeris is available. For this article, I generated the date for Detroit, Michigan, on January 15, 2002.
s = DOP * sO, which means
Positioning Accuracy =
Dilution of Precision * Measurement Precision.
As an example, let’s say the measurement accuracy is 1 meter. If DOP is 1, then the positioning accuracy is 1 meter. What happens when the DOP is 5? The best positioning accuracy is 5 meters. For best results, always do mission planning to avoid times of high DOP.