Dilution of precision

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.

right|thumb|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.

Introduction

The concept of dilution of precision (DOP) originated with users of the Loran-C navigation system. The idea of geometric DOP is to state how errors in the measurement will affect the final state estimation. This can be defined as:

<math display="block">\operatorname{GDOP} = \frac{\Delta ( \text{output location} )}{\Delta ( \text{measured data} )}</math>

Conceptually you can geometrically imagine errors on a measurement resulting in the <math>\Delta ( \text{measured data} )</math> term changing. Ideally small changes in the measured data will not result in large changes in output location. 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 and tropospheric 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., <math>x</math>, <math>y</math>, <math>z</math>, and <math>t</math>). 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. Consider two overlapping rings, or annuli, of different centres. If they overlap at right angles, the greatest extent of the overlap is much smaller than if they overlap in near parallel. Thus a low DOP value represents a better positional precision due to the wider angular separation between the satellites used to calculate a unit's position. 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:

; HDOP: Horizontal dilution of precision

; VDOP: Vertical dilution of precision

; PDOP: Position (3D) dilution of precision

; TDOP: Time dilution of precision

; GDOP: Geometric dilution of precision

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.

The effect of geometry of the satellites on position error is called geometric dilution of precision (GDOP) and it is roughly interpreted as ratio of position error to the range error. Imagine that a square pyramid is formed by lines joining four satellites with the receiver at the tip of the pyramid. The larger the volume of the pyramid, the better (lower) the value of GDOP; the smaller its volume, the worse (higher) the value of GDOP will be. Similarly, the greater the number of satellites, the better the value of GDOP.

Interpretation

{| class="wikitable"

! DOP Value

! Rating

! Description

| < 1

| Ideal

| Highest possible confidence level to be used for applications demanding the highest possible precision at all times.

| 1–2

| Excellent

| At this confidence level, positional measurements are considered accurate enough to meet all but the most sensitive applications.

| 2–5

| Good

| Represents a level that marks the minimum appropriate for making accurate decisions. Positional measurements could be used to make reliable in-route navigation suggestions to the user.

| 5–10

| Moderate

| Positional measurements could be used for calculations, but the fix quality could still be improved. A more open view of the sky is recommended.

| 10–20

| Fair

| Represents a low confidence level. Positional measurements should be discarded or used only to indicate a very rough estimate of the current location.

| > 20

| Poor

| At this level, measurements should be discarded.

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.

Computation

As a first step in computing DOP,

<math display="block">\begin{align}

\operatorname{PDOP} &= \sqrt{\sigma_x^2 + \sigma_y^2 + \sigma_z^2}\\

\operatorname{TDOP} &= \sqrt{\sigma_{t}^2}\\

\operatorname{GDOP} &= \sqrt{\operatorname{PDOP}^2 + \operatorname{TDOP}^2}\\

&= \sqrt{\operatorname{tr} Q}\\

\end{align}</math>

Notice GDOP is the square root of the trace of the <math>Q</math> matrix.

The horizontal and vertical dilution of precision,

<math display="block">\begin{align}

\operatorname{HDOP} &= \sqrt{\sigma_n^2 + \sigma_e^2} \\

\operatorname{VDOP} &= \sqrt{\sigma_u^2}

\end{align},</math>

are both dependent on the coordinate system used. To correspond to the local east-north-up coordinate system,

EDOP^2 x x x

x NDOP^2 x x

x x VDOP^2 x

x x x TDOP^2

and the derived dilutions:

<math display="block">\begin{align}

\operatorname{GDOP} &= \sqrt{\operatorname{EDOP}^2 + \operatorname{NDOP}^2 + \operatorname{VDOP}^2 + \operatorname{TDOP}^2} \\

\operatorname{HDOP} &= \sqrt{\operatorname{EDOP}^2 + \operatorname{NDOP}^2} \\

\operatorname{PDOP} &= \sqrt{\operatorname{EDOP}^2 + \operatorname{NDOP}^2 + \operatorname{VDOP}^2}

\end{align}</math>

Dilution of precision

Introduction

Interpretation

Computation

See also

References

Further reading