Dilution of precision (navigation)

Dilution of precision (DOP), or geometric dilution of precision (GDOP), is a term used in satellite navigation and geomatics engineering to specify the error propagation as a mathematical effect of navigation satellite geometry on positional measurement precision.

The concept of dilution of precision (DOP) originated with users of the Loran-C navigation system.

This can be defined as:[2] Conceptually you can geometrically imagine errors on a measurement resulting in the

The opposite of this ideal is the situation where the solution is very sensitive to measurement errors.

The interpretation of this formula is shown in the figure to the right, showing two possible scenarios with acceptable and poor GDOP.

With the wide adoption of satellite navigation systems, the term has come into much wider usage.

Neglecting ionospheric [3] and tropospheric[4] effects, the signal from navigation satellites has a fixed precision.

Therefore, the relative satellite-receiver geometry plays a major role in determining the precision of estimated positions and times.

Due to the relative geometry of any given satellite to a receiver, the precision in the pseudorange of the satellite translates to a corresponding component in each of the four dimensions of position measured by the receiver (i.e.,

The precision of multiple satellites in view of a receiver combine according to the relative position of the satellites to determine the level of precision in each dimension of the receiver measurement.

When visible navigation satellites are close together in the sky, the geometry is said to be weak and the DOP value is high; when far apart, the geometry is strong and the DOP value is low.

Other factors that can increase the effective DOP are obstructions such as nearby mountains or buildings.

DOP can be expressed as a number of separate measurements: These values follow mathematically from the positions of the usable satellites.

Signal receivers allow the display of these positions (skyplot) as well as the DOP values.

The term can also be applied to other location systems that employ several geographical spaced sites.

It can occur in electronic-counter-counter-measures (electronic warfare) when computing the location of enemy emitters (radar jammers and radio communications devices).

Using such an interferometry technique can provide certain geometric layout where there are degrees of freedom that cannot be accounted for due to inadequate configurations.

Similarly, the greater the number of satellites, the better the value of GDOP.

The DOP factors are functions of the diagonal elements of the covariance matrix of the parameters, expressed either in a global or a local geodetic frame.

As a first step in computing DOP,[5] consider the unit vectors from the receiver to satellite

denote the position of satellite i. Formulate the matrix, A, which (for 4 pseudorange measurement residual equations) is: The first three elements of each row of A are the components of a unit vector from the receiver to the indicated satellite.

The last element of each row refers to the partial derivative of pseudorange w.r.t.

For the preceding case of 4 range measurement residual equations:

is a noise covariance matrix rather than the noise correlation matrix used in DOP, and the reason DOP makes this substitution is to obtain a relative error.

theory to be strictly applicable, either the input noise distributions need to be Gaussian or the measurement noise standard deviations need to be small relative to rate of change in the output near the solution.

This (i.e. for the 4 time of arrival/range measurement residual equations) computation is in accordance with [6] where the weighting matrix,

In other cases, for example when trying to locate someone broadcasting on an international distress frequency,

(Regarding "in place of the TDOP component": Since the clocks on the legacy International Cospas-Sarsat Programme LEO satellites are much less accurate than GPS clocks, discarding their time measurements would actually increase the geolocation solution accuracy.)

The horizontal and vertical dilution of precision, are both dependent on the coordinate system used.

To correspond to the local east-north-up coordinate system, and the derived dilutions:

Understanding dilution of precision with an example. In A someone has measured the distance to two landmarks, and plotted their point as the intersection of two circles with the measured radius. In B the measurement has some error bounds, and their true location will lie anywhere in the green area. In C the measurement error is the same, but the error on their position has grown considerably due to the arrangement of the landmarks.